A386413 G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^2)^(2/3).
1, 6, 63, 792, 10935, 160056, 2438667, 38263752, 614014830, 10029572280, 166203389781, 2787232297680, 47213065271268, 806618756189736, 13883029872725475, 240491818267745760, 4189678646994012501, 73357895462268102840, 1290223574267814268290, 22784365638084466567800
Offset: 0
Links
- Paolo Xausa, Table of n, a(n) for n = 0..750
- Wikipedia, Fuss-Catalan number
Programs
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Mathematica
A386413[n_] := 9^n*Binomial[(4*n + 2)/3, n]/(2*n + 1); Array[A386413, 25, 0] (* Paolo Xausa, Aug 01 2025 *)
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PARI
apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r); a(n) = 9^n*apr(n, 4/3, 2/3);
Formula
a(n) = 9^n * binomial((4*n+2)/3,n)/(2*n+1).
G.f.: B(x)^2, where B(x) is the g.f. of A078532.
D-finite with recurrence n*(n-2)*(n+2)*a(n) -216*(2*n-5)*(4*n-7)*(4*n-1)*a(n-3)=0. - R. J. Mathar, Jul 30 2025