cp's OEIS Frontend

A370101 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(4*n-k-1,n-k).

%I A370101 #27 Aug 09 2025 07:33:11
%S A370101 1,7,97,1519,25089,427007,7408897,130287871,2313945089,41409732607,
%T A370101 745530884097,13488086405119,245014271688705,4465915098890239,
%U A370101 81637668328243201,1496095489290731519,27477504726883368961,505627095685486608383,9320167322334416338945
%N A370101 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(4*n-k-1,n-k).
%F A370101 a(n) = [x^n] ( (1+x)^4/(1-x)^3 )^n.
%F A370101 The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1-x)^3/(1+x)^4 ). See A365846.
%F A370101 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n,k). - _Seiichi Manyama_, Jul 31 2025
%F A370101 a(n) ~ 2^(9*n + 3/2) / (7 * sqrt(Pi*n) * 3^(3*n - 1/2)). - _Vaclav Kotesovec_, Jul 31 2025
%F A370101 a(n) = Sum_{k=0..n} 2^k * binomial(3*n+k-1,k). - _Seiichi Manyama_, Aug 01 2025
%F A370101 a(n) = [x^n] 1/((1-x) * (1-2*x)^(3*n)). - _Seiichi Manyama_, Aug 09 2025
%t A370101 Table[Sum[Binomial[4n,k]Binomial[4n-k-1,n-k],{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Dec 09 2024 *)
%t A370101 Table[Sum[2^k*(-1)^(n-k)*Binomial[4*n, k], {k, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Jul 31 2025 *)
%o A370101 (PARI) a(n) = sum(k=0, n, binomial(4*n, k)*binomial(4*n-k-1, n-k));
%Y A370101 Cf. A001448, A370100, A370102.
%Y A370101 Cf. A119259, A370097.
%Y A370101 Cf. A365846.
%K A370101 nonn
%O A370101 0,2
%A A370101 _Seiichi Manyama_, Feb 10 2024