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 A383274 #43 May 07 2025 11:33:10
%S A383274 1,13,441,20629,1119361,66116013,4126228569,267666251733,
%T A383274 17868312820737,1219477111897933,84701899713767161,
%U A383274 5967906378862013973,425503428034568158081,30642774518964618986989,2225692868157573335052441,162858794856607965831417429,11993850186156155815298686977
%N A383274 a(n) = Sum_{i,j = 0..n} C(n, i)^2*C(n, j)^2*C(i+j, i)*2^(i+j).
%C A383274 Conjecture 1: Let S(p) = Sum_{0<k<p} a(k) with p a prime.
%C A383274 If p == 1,9 (mod 20) with p = x^2 + 5*y^2, then S(p) == 4*x^2-2*p (mod p^2).
%C A383274 If p == 3,7 (mod 20) with 2*p = x^2 + 5*y^2, then S(p) == 2*x^2 - 2*p (mod p^2).
%C A383274 If p == 11, 13, 17, 19 (mod 20), then S(p) == 0 (mod p^2).
%C A383274 Conjecture 2: For any prime p == 1,3,7,9 (mod 20), we have Sum_{0<k<p}(8k+5)*a(k) == 14/5*p (mod p^2).
%C A383274 Both conjectures have been verified for odd primes smaller than 1000.
%H A383274 Zhi-Wei Sun, <a href="/A383274/b383274.txt">Table of n, a(n) for n = 0..85</a>
%H A383274 Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/491655">On a_n(x) = Sum_{i,j=0..n}C(n,i)^2*C(n,j)^2*C(i+j,i)*x^(i+j) (I)</a>, Question 491655 at MathOverflow, April 24, 2025.
%H A383274 Zhi-Wei Sun, <a href="https://arxiv.org/abs/2505.02767">A family of polynomials and related congruences and series</a>, arXiv:2505.02767 [math.NT], 2025.
%F A383274 a(n) ~ 3^(4*n+3) / (8 * sqrt(7) * Pi^(3/2) * n^(3/2)). - _Vaclav Kotesovec_, Apr 27 2025
%e A383274 a(1) = 13 since Sum_{i,j = 0,1}C(1,i)^2*C(1,j)^2*C(i+j,i)^2*2^(i+j) = Sum_{i,j = 0,1} C(i+j,i)^2*2^(i+j) = 2^0 + 2^1 + 2^1 + C(2,1)*2^2 = 13.
%p A383274 a := proc(n) option remember; local i, j; add(add(binomial(n, i)^2 * binomial(n, j)^2 * binomial(i+j, i) * 2^(i+j), i = 0..n), j = 0..n) end: seq(a(n), n=0..16); # _Peter Luschny_, Apr 27 2025
%t A383274 a[n_] := a[n] = Sum[Binomial[n,i]^2*Binomial[n,j]^2*Binomial[i+j,i]*2^(i+j), {i,0,n}, {j,0,n}]; Table[a[n], {n,0,17}]
%o A383274 (Python)
%o A383274 from math import comb
%o A383274 def A383274(n): return sum((comb(n,i)**2<<i)*sum(comb(n,j)**2*comb(i+j,i)<<j for j in range(n+1)) for i in range(n+1)) # _Chai Wah Wu_, Apr 27 2025
%Y A383274 Cf. A005259.
%K A383274 nonn
%O A383274 0,2
%A A383274 _Zhi-Wei Sun_, Apr 26 2025