A386768 a(n) = Sum_{k=0..n} 5^k * 2^(n-k) * binomial(3*n+k-1,k).
1, 17, 589, 23063, 952421, 40527732, 1758058219, 77293019898, 3431959098741, 153547092814172, 6911193017626324, 312596792782451183, 14196172772254858211, 646897139198653660412, 29563753017571135649154, 1354477988702509748029668, 62191803671962046948722581
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, 5^k*2^(n-k)*binomial(3*n+k-1, k));
Formula
a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k).
a(n) = [x^n] ( (1+2*x)^4/(1-3*x)^3 )^n.
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.
a(n) = Sum_{k=0..n} 5^k * (-3)^(n-k) * binomial(4*n,k).
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