cp's OEIS Frontend

A364510 a(n) = binomial(4*n, n)^2.

%I A364510 #18 Oct 06 2023 10:41:34
%S A364510 1,16,784,48400,3312400,240374016,18116083216,1401950721600,
%T A364510 110634634890000,8862957169158400,718528370729238784,
%U A364510 58818762721626513424,4853704694918904043024,403242220875862752160000,33694913171561404510440000,2829611125043050701300998400
%N A364510 a(n) = binomial(4*n, n)^2.
%H A364510 Paolo Xausa, <a href="/A364510/b364510.txt">Table of n, a(n) for n = 0..500</a>
%H A364510 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dixon%27s_identity">Dixon's identity</a>
%F A364510 a(n) = Sum_{i = -n..n} (-1)^i * binomial(2*n, n+i)^2 * binomial(4*n, 2*n+i).
%F A364510 Compare with Dixon's identity: Sum_{i = -n..n} (-1)^i * binomial(2*n, n+i)^3 = (3*n)!/n!^3.
%F A364510 a(n) = A005810(n)^2.
%F A364510 P-recursive: a(n) = 16 * ( (4*n - 1)*(4*n - 2)*(4*n - 3)/(3*n*(3*n - 1)*(3*n - 2)) )^2 * a(n-1) with a(0) = 1.
%F A364510 a(n) ~ c^n * 2/(3*Pi*n), where c = (2^16)/(3^6).
%F A364510 a(n) = [x^n] G(x)^(16*n), where the power series G(x) = 1 + x + 9*x^2 + 225*x^3 + 7525*x^4 + 295228*x^5 + 12787152*x^6 + 592477457*x^7 + 28827755219*x^8 + ... appears to have integer coefficients.
%F A364510 exp( Sum_{n > = 1} a(n)*x^n/n ) = F(x)^16, where the power series F(x) =  1 + x + 25*x^2 + 1033*x^3 + 53077*x^4 + 3081944*x^5 + 193543624*x^6 + 12835533333*x^7 + 886092805699*x^8 + ... appears to have integer coefficients.
%F A364510 The supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3^r)) hold for all primes p >= 5 and all positive integers n and r.
%F A364510 a(n) = [x^(3*n)] ( (1 - x)^(6*n)*Legendre_P(2*n, (1 + x)/(1 - x)) ). - _Peter Bala_, Aug 14 2023
%p A364510 seq( binomial(4*n,n)^2, n = 0..15);
%t A364510 A364510[n_]:=Binomial[4n,n]^2;Array[A364510,15,0] (* _Paolo Xausa_, Oct 05 2023 *)
%Y A364510 Cf. A002894, A005810, A364506, A364511, A364512.
%Y A364510 Row 2 of A364509.
%K A364510 nonn,easy
%O A364510 0,2
%A A364510 _Peter Bala_, Jul 28 2023