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 A364507 #25 Oct 07 2023 06:58:36
%S A364507 1,40,5880,1101100,229265400,50678855040,11641642112100,
%T A364507 2746924727976000,661097260785195000,161538994454795003200,
%U A364507 39949572934939198410880,9976687616280042928424700,2511716999955421326631644900,636662322699394050738883008000
%N A364507 a(n) = (5*n)!*(4*n)! / ((3*n)!^2 * (2*n)! * n!).
%C A364507 Row 2 of A364506.
%H A364507 Paolo Xausa, <a href="/A364507/b364507.txt">Table of n, a(n) for n = 0..400</a>
%H A364507 Romeo Meštrović, <a href="https://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)</a>, arXiv:1111.3057 [math.NT], 2011.
%H A364507 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dixon%27s_identity">Dixon's identity</a>
%F A364507 a(n) = Sum_{k = -n..n} (-1)^k * binomial(2*n, n + k) * binomial(4*n, 2*n + k)^2 (showing a(n) to be integral). Compare with Dixon's identity, Sum_{k = -n..n} (-1)^k * binomial(2*n, n + k)^3 = (3*n)!/n!^3 = A006480(n).
%F A364507 P-recursive: a(n) = (20/9)*(4*n-1)*(4*n-3)*(5*n-1)*(5*n-2)*(5*n-3)*(5*n-4)/((3*n-1)^2*(3*n-2)^2*n^2) * a(n-1) with a(0) = 1.
%F A364507 a(n) ~ c^n * sqrt(10)/(6*Pi*n), where c = (2^6)*(5^5)/(3^6).
%F A364507 a(n) = [x^n] G(x)^(20*n), where the power series G(x) = 1 + 2*x + 69*x^2 + 5647*x^3 + 618860*x^4 + 79241349*x^5 + 11177111981*x^6 + 1684171189810*x^7 + 266238907746252*x^8 + ... appears to have integer coefficients.
%F A364507 exp( Sum_{n > = 1} a(n)*x^n/n ) = F(x)^20, where the power series F(x) = 1 + 2*x + 149*x^2 + 18647*x^3 + 2913620*x^4 + 515276389*x^5 + 98628630997*x^6 + 19944410220744*x^7 + 4199273746072180*x^8 + ... appears to have integer coefficients.
%F A364507 Conjecture: 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 [added Aug 04 2023: the conjecture follows from Meštrović. equation 39].
%F A364507 a(n) = binomial(4*n,n)*binomial(5*n,2*n). - _Christian Krause_, Aug 03 2023
%e A364507 Examples of supercongruences:
%e A364507 a(7) - a(1) = 2746924727976000 - 40 = (2^3)*5*(7^4)*28601881799 == 0 (mod 7^4).
%e A364507 a(11) - a(1) = 9976687616280042928424700 - 40 = (2^2)*5*(11^3)*18397*3568463* 5708869513 == 0 (mod 11^3).
%p A364507 seq( (5*n)!*(4*n)!*(2*n)! / ((3*n)!^2 * (2*n)!^2 * n!), n = 0..15);
%t A364507 A364507[n_]:=(5n)!(4n)!/((3n)!^2(2n)!n!);Array[A364507,15,0] (* _Paolo Xausa_, Oct 06 2023 *)
%Y A364507 Cf. A001450, A005810, A006480, A364506, A364508.
%K A364507 nonn,easy
%O A364507 0,2
%A A364507 _Peter Bala_, Jul 27 2023