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 A386768 #16 Aug 03 2025 11:16:18
%S A386768 1,17,589,23063,952421,40527732,1758058219,77293019898,3431959098741,
%T A386768 153547092814172,6911193017626324,312596792782451183,
%U A386768 14196172772254858211,646897139198653660412,29563753017571135649154,1354477988702509748029668,62191803671962046948722581
%N A386768 a(n) = Sum_{k=0..n} 5^k * 2^(n-k) * binomial(3*n+k-1,k).
%F A386768 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k).
%F A386768 a(n) = [x^n] ( (1+2*x)^4/(1-3*x)^3 )^n.
%F A386768 The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-3*x)^3 / (1+2*x)^4 ). See A386774.
%F A386768 a(n) = Sum_{k=0..n} 5^k * (-3)^(n-k) * binomial(4*n,k).
%F A386768 D-finite with recurrence +6561*n*(3*n-1)*(866874441*n -1379250238)*(3*n-2)*a(n) +648*(-3285736631046*n^4 +7965087872184*n^3 -621409760406*n^2 -9688518250831*n +5867806764110)*a(n-1) +1920*(-7264105318332*n^4 +60745334410890*n^3 -195779508237450*n^2 +280383483469585*n -148039402286753)*a(n-2) -51200*(2*n-5)*(4*n-9) *(4581663714*n-6698674013)*(4*n-11)*a(n-3)=0. - _R. J. Mathar_, Aug 03 2025
%o A386768 (PARI) a(n) = sum(k=0, n, 5^k*2^(n-k)*binomial(3*n+k-1, k));
%Y A386768 Cf. A386766, A386767.
%Y A386768 Cf. A386774.
%K A386768 nonn
%O A386768 0,2
%A A386768 _Seiichi Manyama_, Aug 02 2025