# Project Euler 638 - Weighted Lattice Paths
# sum_{k=1..7} [2*10^k+k choose 10^k+k]_q (mod 1e9+7)
# Gaussian binomial coefficient via product formula.
import euler.nt { mod_pow }
const MOD: i64 = 1000000007
# Gaussian binomial coefficient [m choose n]_q mod `mod`.
function gaussian_binom(m: i64, n_in: i64, q: i64, mod: i64) -> i64 {
let mut n: i64 = n_in
if n < 0 || n > m {
return 0
}
if n == 0 || n == m {
return 1
}
if q % mod == 1 {
# regular binomial
if n > m - n {
n = m - n
}
if n == 0 {
return 1
}
let mut num: i64 = 1
let mut den: i64 = 1
let start: i64 = m - n
let mut i: i64 = 1
while i <= n {
num = ((num as i128) * ((start + i) % mod) % mod) as i64
den = ((den as i128) * (i % mod) % mod) as i64
i = i + 1
}
return ((num as i128) * mod_pow(den, mod - 2, mod) % mod) as i64
}
if n > m - n {
n = m - n
}
if n == 0 {
return 1
}
let b: i64 = m - n
let mut num_prod: i64 = 1
let mut den_prod: i64 = 1
let mut qmod: i64 = q % mod
if qmod < 0 {
qmod = qmod + mod
}
let mut pow_den: i64 = qmod
let mut pow_num: i64 = mod_pow(qmod, b + 1, mod)
let mut i: i64 = 0
while i < n {
num_prod = ((num_prod as i128) * (mod + 1 - pow_num) % mod) as i64
den_prod = ((den_prod as i128) * (mod + 1 - pow_den) % mod) as i64
pow_den = ((pow_den as i128) * qmod % mod) as i64
pow_num = ((pow_num as i128) * qmod % mod) as i64
i = i + 1
}
return ((num_prod as i128) * mod_pow(den_prod, mod - 2, mod) % mod) as i64
}
function main() -> i32 {
let mod: i64 = MOD
let mut ans: i64 = 0
let mut k: i32 = 1
while k <= 7 {
let mut n: i64 = 1
let mut j: i32 = 0
while j < k {
n = n * 10
j = j + 1
}
n = n + (k as i64)
ans = (ans + gaussian_binom(2 * n, n, (k as i64), mod)) % mod
k = k + 1
}
printf("%lld\n", ans)
return 0
}
Generated C
#include <stdint.h>
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
/* Flow runtime helpers */
typedef struct flow_temp_node { struct flow_temp_node* next; } flow_temp_node;
static flow_temp_node* flow_temp_head = NULL;
static int flow_temp_atexit_set = 0;
__attribute__((unused)) static void flow_temp_free_all(void) {
while (flow_temp_head) {
flow_temp_node* n = flow_temp_head;
flow_temp_head = n->next;
free(n);
}
}
__attribute__((unused)) static void* flow_temp_alloc(size_t nbytes) {
flow_temp_node* node = (flow_temp_node*)malloc(sizeof(flow_temp_node) + nbytes);
if (!node) return NULL;
node->next = flow_temp_head;
flow_temp_head = node;
if (!flow_temp_atexit_set) {
flow_temp_atexit_set = 1;
atexit(flow_temp_free_all);
}
return (void*)(node + 1);
}
#ifndef FLOW_DIAG
#define FLOW_DIAG(msg) fprintf(stderr, "%s", (msg))
#endif
#ifndef FLOW_LOG
#define FLOW_LOG(fmt, ...) printf(fmt, __VA_ARGS__)
#endif
#ifndef FLOW_LOG_EMPTY
#define FLOW_LOG_EMPTY(fmt) printf(fmt)
#endif
static char* flow_strcat(const char* a, const char* b) {
size_t la = strlen(a ? a : ""), lb = strlen(b ? b : "");
char* r = (char*)flow_temp_alloc(la + lb + 1);
if (!r) return NULL;
if (la) memcpy(r, a, la);
if (lb) memcpy(r + la, b, lb);
r[la + lb] = '\0';
return r;
}
#define __flow_in_arr(arr, val) __extension__ ({ \
int _found = 0; \
size_t _n = sizeof(arr)/sizeof((arr)[0]); \
for (size_t _i = 0; _i < _n; _i++) { \
if ((arr)[_i] == (val)) { _found = 1; break; } \
} _found; })
/* Unified fault handler (MISRA #279) — override with -DFLOW_FAULT_HANDLER=fn */
#ifndef FLOW_FAULT_HANDLER
__attribute__((unused)) static inline void flow_fault_handler(const char* msg) {
fprintf(stderr, "flow: %s\n", msg ? msg : "fault");
abort();
#if defined(__GNUC__) || defined(__clang__)
__builtin_unreachable();
#endif
}
#else
#define flow_fault_handler FLOW_FAULT_HANDLER
#endif
#define flow_div_by_zero_handler() flow_fault_handler("division by zero")
#define flow_shift_ub_handler() flow_fault_handler("invalid shift (amount out of range or left-shift of negative)")
#ifndef FLOW_CHECKED_DIV
#define FLOW_CHECKED_DIV(L, R) (((R) != 0) ? ((L) / (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_MOD
#define FLOW_CHECKED_MOD(L, R) (((R) != 0) ? ((L) % (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHL
#define FLOW_CHECKED_SHL(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull)) && ((L) >= 0)) ? ((L) << (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHR
#define FLOW_CHECKED_SHR(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull))) ? ((L) >> (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#include <math.h>
void* _ui_state = NULL;
static inline float i32_to_f32(int32_t v) { return (float)v; }
/* Host stub for @gpu kernels (device codegen replaces this). */
static inline int32_t gpu_thread_id(void) { return 0; }
int64_t gcd_i64_i64(int64_t a0, int64_t b0);
int64_t lcm_i64_i64(int64_t a, int64_t b);
int64_t isqrt_i64(int64_t n);
int64_t mulmod_i64_i64_i64(int64_t a0, int64_t b0, int64_t mod);
int64_t mod_pow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod);
bool is_prime_i64(int64_t n);
int64_t gaussian_binom_i64_i64_i64_i64(int64_t m, int64_t n_in, int64_t q, int64_t mod);
int32_t main(void);
static const int64_t MOD = 1000000007;
int64_t gcd_i64_i64(int64_t a0, int64_t b0) {
int64_t a = a0;
int64_t b = b0;
while (b != 0) {
int64_t t = FLOW_CHECKED_MOD((a), (b));
a = b;
b = t;
}
return a;
}
int64_t lcm_i64_i64(int64_t a, int64_t b) {
if ((a == 0 || b == 0)) {
return 0;
}
return (FLOW_CHECKED_DIV((a), (gcd_i64_i64(a, b))) * b);
}
int64_t isqrt_i64(int64_t n) {
if (n < 2) {
return n;
}
int64_t x = n;
int64_t y = FLOW_CHECKED_DIV(((x + 1)), (2));
while (y < x) {
x = y;
y = FLOW_CHECKED_DIV(((x + FLOW_CHECKED_DIV((n), (x)))), (2));
}
return x;
}
int64_t mulmod_i64_i64_i64(int64_t a0, int64_t b0, int64_t mod) {
int64_t a = FLOW_CHECKED_MOD((a0), (mod));
int64_t b = FLOW_CHECKED_MOD((b0), (mod));
int64_t result = 0;
while (b > 0) {
if (FLOW_CHECKED_MOD((b), (2)) == 1) {
result = FLOW_CHECKED_MOD(((result + a)), (mod));
}
a = FLOW_CHECKED_MOD(((a * 2)), (mod));
b = FLOW_CHECKED_DIV((b), (2));
}
return result;
}
int64_t mod_pow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod) {
if (mod == 1) {
return 0;
}
int64_t result = 1;
int64_t b = FLOW_CHECKED_MOD((base), (mod));
int64_t e = exp;
while (e > 0) {
if (FLOW_CHECKED_MOD((e), (2)) == 1) {
result = mulmod_i64_i64_i64(result, b, mod);
}
b = mulmod_i64_i64_i64(b, b, mod);
e = FLOW_CHECKED_DIV((e), (2));
}
return result;
}
bool is_prime_i64(int64_t n) {
if (n < 2) {
return 0;
}
if (n < 4) {
return 1;
}
if ((FLOW_CHECKED_MOD((n), (2)) == 0 || FLOW_CHECKED_MOD((n), (3)) == 0)) {
return 0;
}
int64_t i = 5;
while ((i * i) <= n) {
if ((FLOW_CHECKED_MOD((n), (i)) == 0 || FLOW_CHECKED_MOD((n), ((i + 2))) == 0)) {
return 0;
}
i = (i + 6);
}
return 1;
}
int64_t gaussian_binom_i64_i64_i64_i64(int64_t m, int64_t n_in, int64_t q, int64_t mod) {
int64_t n = n_in;
if ((n < 0 || n > m)) {
return 0;
}
if ((n == 0 || n == m)) {
return 1;
}
if (FLOW_CHECKED_MOD((q), (mod)) == 1) {
if (n > (m - n)) {
n = (m - n);
}
if (n == 0) {
return 1;
}
int64_t num = 1;
int64_t den = 1;
int64_t start = (m - n);
int64_t i = 1;
while (i <= n) {
num = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(num)) * FLOW_CHECKED_MOD(((start + i)), (mod)))), (mod))));
den = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(den)) * FLOW_CHECKED_MOD((i), (mod)))), (mod))));
i = (i + 1);
}
return ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(num)) * mod_pow_i64_i64_i64(den, (mod - 2), mod))), (mod))));
}
if (n > (m - n)) {
n = (m - n);
}
if (n == 0) {
return 1;
}
int64_t b = (m - n);
int64_t num_prod = 1;
int64_t den_prod = 1;
int64_t qmod = FLOW_CHECKED_MOD((q), (mod));
if (qmod < 0) {
qmod = (qmod + mod);
}
int64_t pow_den = qmod;
int64_t pow_num = mod_pow_i64_i64_i64(qmod, (b + 1), mod);
int64_t i = 0;
while (i < n) {
num_prod = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(num_prod)) * ((mod + 1) - pow_num))), (mod))));
den_prod = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(den_prod)) * ((mod + 1) - pow_den))), (mod))));
pow_den = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(pow_den)) * qmod)), (mod))));
pow_num = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(pow_num)) * qmod)), (mod))));
i = (i + 1);
}
return ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(num_prod)) * mod_pow_i64_i64_i64(den_prod, (mod - 2), mod))), (mod))));
}
int32_t main(void) {
int64_t mod = MOD;
int64_t ans = 0;
int32_t k = 1;
while (k <= 7) {
int64_t n = 1;
int32_t j = 0;
while (j < k) {
n = (n * 10);
j = (j + 1);
}
n = (n + ((int64_t)(k)));
ans = FLOW_CHECKED_MOD(((ans + gaussian_binom_i64_i64_i64_i64((2 * n), n, ((int64_t)(k)), mod))), (mod));
k = (k + 1);
}
printf("%lld\n", ans);
return 0;
}