A386896 a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(3*n-k,n-k).
1, 9, 125, 1932, 31365, 523809, 8910356, 153544680, 2671398309, 46822319115, 825501663525, 14623742203200, 260088366645900, 4641248247561324, 83059406374007720, 1490097583932329232, 26790218420643034533, 482571492068274975135, 8707190579448431827991
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, binomial(5*n+1, k)*binomial(3*n-k, n-k));
Formula
a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(2*n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x))^(2*n+1).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(3*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(2*n+k,k) * binomial(3*n-k,n-k).
a(n) = binomial(3*n, n)*hypergeom([-1-5*n, -n], [-3*n], -1). - Stefano Spezia, Aug 07 2025
D-finite with recurrence 202*n*(n-1)*(2*n-1)*(2*n-3)*a(n) -3*(n-1)*(2*n-3) *(14093*n^2-15245*n+5226)*a(n-1) +4*(355081*n^4 -1597876*n^3 +2789549*n^2 -2405270*n+926160)*a(n-2) -3840*(5*n-11)*(5*n-9) *(5*n-13)*(5*n-12)*a(n-3)=0. - R. J. Mathar, Aug 21 2025