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 A357565 #10 Oct 25 2022 05:16:01
%S A357565 5,10,114,2926,109106,4846260,234488526,11913003294,625130924082,
%T A357565 33590792825200,1838547540484364,102135528447552060,
%U A357565 5743779960435245774,326352202770939600460,18706076476872783254286,1080345839256279791104926,62806507721442655949609010
%N A357565 a(n) = 3*Sum_{k = 0..n} binomial(n+k-1,k)^2 + 2*Sum_{k = 0..n} binomial(n+k-1,k)^3.
%C A357565 Conjectures:
%C A357565 1) a(p) == a(1) (mod p^5) for all odd primes p except p = 5 (checked up to p = 271).
%C A357565 2) a(p^r) == a(p^(r-1)) (mod p^(3*r+3)) for r >= 2 and all primes p >= 3.
%C A357565 3) More generally, let m be a positive integer and set u(n) = (m + 2)*Sum_{k = 0..m*n} binomial(n+k-1,k)^2 + 2*m*Sum_{k = 0..m*n} binomial(n+k-1,k)^3. Then the supercongruences u(p) == u(1) (mod p^5) hold for all primes p >= 7.
%C A357565 4) u(p^r) == u(p^(r-1)) (mod p^(3*r+3)) for r >= 2 and all primes p >= 3.
%e A357565 a(11) - a(1) = 102135528447552060 - 10 = 2*(5^2)*(11^5)*14657* 865363 == 0 (mod 11^5).
%e A357565 a(5^2) - a(5) = 581553752659150682384860284864053981408760 - 4846260 = 3*(2^2)*(5^9)*5611847956825027*4421531072180960789 == 0 (mod 5^9)
%p A357565 seq(add( 3*binomial(n+k-1,k)^2 + 2*binomial(n+k-1,k)^3, k = 0..n ), n = 0..20);
%o A357565 (PARI) a(n) = 3*sum(k = 0, n, binomial(n+k-1,k)^2) + 2*sum(k = 0, n, binomial(n+k-1,k)^3); \\ _Michel Marcus_, Oct 25 2022
%Y A357565 Cf. A357566, A357671, A357672, A357673, A357674.
%K A357565 nonn,easy
%O A357565 0,1
%A A357565 _Peter Bala_, Oct 16 2022