This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357566 #14 May 31 2025 03:56:51
%S A357566 1,32,3556224,4816142496896,14260946236464636800,
%T A357566 62923492736113950202540032,355372959542696519903013302282592,
%U A357566 2376354966106399942850054560101358877184,17973185649572984869873798116070605084766512000,149319509846904520286037745483655872001727895961600000
%N 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.
%C A357566 Conjectures:
%C A357566 1) a(p) == a(1) (mod p^5) for all odd primes p except p = 5 (checked up to p = 271).
%C A357566 2) a(p^r) == a(p^(r-1)) (mod p^(3*r+3)) for r >= 2 and all primes p >= 5.
%C A357566 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.
%C A357566 4) u(p^r) == u(p^(r-1)) (mod p^(3*r+3)) for r >= 2 and all primes p >= 5.
%F A357566 a(n) ~ 2^(24*n) / (1323*Pi^6*n^6). - _Vaclav Kotesovec_, May 31 2025
%e A357566 a(7) - a(1) = 2376354966106399942850054560101358877184 - 32 = (2^5)*(7^5)*19*31*317*339247*25170329*2771351868561767 == 0 (mod 7^5).
%p A357566 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);
%t A357566 Table[Sum[Binomial[n+k-1,k]^2, {k, 0, n}]^3 * Sum[Binomial[n+k-1,k]^3, {k, 0, n}]^2, {n, 0, 10}] (* _Vaclav Kotesovec_, May 31 2025 *)
%o A357566 (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
%Y A357566 Cf. A357565, A357671, A357672, A357673, A357674.
%Y A357566 Cf. A333592.
%K A357566 nonn,easy
%O A357566 0,2
%A A357566 _Peter Bala_, Oct 16 2022