A357566 a(n) = ( Sum_{k = 0..n} binomial(n+k-1,k)^2 )^3 * ( Sum_{k = 0..n} binomial(n+k-1,k)^3 )^2.

1, 32, 3556224, 4816142496896, 14260946236464636800, 62923492736113950202540032, 355372959542696519903013302282592, 2376354966106399942850054560101358877184, 17973185649572984869873798116070605084766512000, 149319509846904520286037745483655872001727895961600000

Offset

0,2

Comments

Conjectures:

1) a(p) == a(1) (mod p^5) for all odd primes p except p = 5 (checked up to p = 271).

2) a(p^r) == a(p^(r-1)) (mod p^(3*r+3)) for r >= 2 and all primes p >= 5.

3) More generally, let m be a positive integer and set u(n) = ( Sum_{k = 0..m*n} binomial(n+k-1,k)^2 )^(m+2) * ( Sum_{k = 0..m*n} binomial(n+k-1,k)^3 )^(2*m). Then the supercongruences u(p) == u(1) (mod p^5) hold for all primes p >= 5.

4) u(p^r) == u(p^(r-1)) (mod p^(3*r+3)) for r >= 2 and all primes p >= 5.

Links

Table of n, a(n) for n=0..9.

Example

a(7) - a(1) = 2376354966106399942850054560101358877184 - 32 = (2^5)*(7^5)*19*31*317*339247*25170329*2771351868561767 == 0 (mod 7^5).

Maple

seq((add(binomial(n+k-1, k)^2, k = 0..n))^3 * (add( binomial(n+k-1, k)^3, k = 0..n))^2, n = 0..20);

Prog

(PARI) a(n) = sum(k = 0, n, binomial(n+k-1, k)^2)^3 * sum(k = 0, n, binomial(n+k-1, k)^3)^2; \\ Michel Marcus, Oct 25 2022

Crossrefs

Cf. A357565, A357671, A357672, A357673, A357674.

Sequence in context: A017549 A217027 A087927 * A265217 A368327 A224087

Adjacent sequences: A357563 A357564 A357565 * A357567 A357568 A357569

Keyword

nonn,easy

Author

Peter Bala, Oct 16 2022

Status

approved