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A370102 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(5*n-k-1,n-k).

%I A370102 #25 Aug 09 2025 07:33:36
%S A370102 1,8,128,2312,44032,864008,17282432,350353928,7172939776,147972367880,
%T A370102 3070951360128,64044689834760,1341056098444288,28176478479561992,
%U A370102 593725756425591680,12542160174109922312,265525958014053580800,5632170795392966388744
%N A370102 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(5*n-k-1,n-k).
%F A370102 a(n) = [x^n] ( (1+x)^4/(1-x)^4 )^n.
%F A370102 The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1-x)^4/(1+x)^4 ). See A365847.
%F A370102 From _Peter Bala_, Jul 20 2024: (Start)
%F A370102 a(n) = binomial(5*n-1, n)*hypergeom([-n, -4*n], [1 - 5*n], -1).
%F A370102 For n >=1, a(n) = (4/3) * [x^n] S(x)^(3*n) = (4/5) * [x^n] (1/S(-x))^(5*n), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the sequence of large Schröder numbers A006318.
%F A370102 n*(4*n - 3)*(2*n - 1)*(4*n - 1)*(85*n^4 - 510*n^3 + 1138*n^2 - 1119*n + 409)*a(n) = 2*(29665*n^8 - 237320*n^7 + 794282*n^6 - 1443212*n^5 + 1544750*n^4 - 987560*n^3 + 363568*n^2 - 69168*n + 5040)*a(n-1) + (n - 2)*(4*n - 7)*(2*n - 3)*(4*n - 5)*(85*n^4 - 170*n^3 + 118*n^2 - 33*n + 3)*a(n-2) with a(0) = 1 and a(1) = 8.
%F A370102 The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
%F A370102 Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and positive integers n and r. (End)
%F A370102 a(n) ~ (349 + 85*sqrt(17))^n / (17^(1/4) * sqrt(Pi*n) * 2^(5*n - 1/2)). - _Vaclav Kotesovec_, Aug 08 2024
%F A370102 From _Seiichi Manyama_, Aug 09 2025: (Start)
%F A370102 a(n) = [x^n] (1-x)^(n-1)/(1-2*x)^(4*n).
%F A370102 a(n) = Sum_{k=0..n} 2^k * binomial(4*n,k) * binomial(n-1,n-k).
%F A370102 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+k-1,k) * binomial(n-1,n-k). (End)
%p A370102 seq(simplify(binomial(5*n-1, n)*hypergeom([-n, -4*n], [1 - 5*n], -1)), n = 0..20); # _Peter Bala_, Jul 29 2024
%o A370102 (PARI) a(n) = sum(k=0, n, binomial(4*n, k)*binomial(5*n-k-1, n-k));
%Y A370102 Cf. A006318, A103885, A365847, A370098, A370099.
%Y A370102 Cf. A001448, A370100, A370101.
%K A370102 nonn,easy
%O A370102 0,2
%A A370102 _Seiichi Manyama_, Feb 10 2024