A386765 a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(3*n+k-1,k).
1, 18, 624, 24432, 1008876, 42927318, 1862060124, 81862383432, 3634739070876, 162615605774568, 7319222860673124, 331046648931192432, 15033834910528707876, 685059700337659528068, 31307482174782491223624, 1434354449577159551751432, 65858845473746133806094876
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, 5^k*3^(n-k)*binomial(3*n+k-1, k));
Formula
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k).
a(n) = [x^n] ( (1+3*x)^4/(1-2*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 * (1-2*x)^3 / (1+3*x)^4 ). See A386771.
a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(4*n,k).
a(n) = (-2)^(-3*n)*81^n - 5^n*binomial(4*n - 1, n)*(hypergeom([1, 4*n], [1+n], 5/3) - 1). - Stefano Spezia, Aug 03 2025