A381826 G.f. A(x) satisfies A(x) = C(x) / (1 - x*A(x)^2), where C(x) is the g.f. of A000108.
1, 2, 8, 41, 241, 1545, 10503, 74429, 543833, 4067510, 30985633, 239560975, 1874831287, 14823253892, 118222204539, 949963236834, 7683289712433, 62499664522578, 510992689465500, 4196824203859773, 34609480384100715, 286461380785102398, 2378954616256505177
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, binomial(2*n+1, k)*binomial(3*n-3*k, n-k))/(2*n+1);
Formula
a(n) = (1/(2*n+1)) * Sum_{k=0..n} binomial(2*n+1,k) * binomial(3*n-3*k,n-k).
D-finite with recurrence 12*n*(3*n+2)*(2*n+1)*(3*n+1)*a(n) +2*(-2365*n^4+2754*n^3-1799*n^2+834*n-144)*a(n-1) +2*(20215*n^4-89442*n^3+158117*n^2-135942*n+47592)*a(n-2) +(-181487*n^4+1469774*n^3-4524589*n^2+6309094*n-3370512)*a(n-3) +124*(n-3)*(2*n-7)*(1797*n^2-9448*n+12568)*a(n-4) -119164*(2*n-7)*(2*n-9)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Mar 10 2025