A396415 Expansion of F_3(x)/x, where F_k(x) is the k-th iteration of x*G3(x) with G3(x) = 1 + x*G3(x)^3.

1, 3, 15, 90, 595, 4184, 30730, 233216, 1816135, 14442004, 116858182, 959568490, 7979302648, 67080255362, 569335836460, 4872950092054, 42019441551195, 364747126728335, 3185048868629538, 27961717170478835, 246666594628933346, 2185543714195145651

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

G.f.: (1/x) * Series_Reversion( H_3(x) ), where H_k(x) is the k-th iteration of x*(1 - x*C(x)) with C(x) = 1 + x*C(x)^2.

a(n) = Sum_{0 = x_0 <= x_1 <= x_2 <= x_3 = n} Product_{k=0..2} (x_k + 1) * binomial(3*x_{k+1} - 2*x_k + 1,x_{k+1} - x_k)/(3*x_{k+1} - 2*x_k + 1).

a(n) = Sum_{k=0..n} (k+1) * binomial(3*n-2*k+1,n-k)/(3*n-2*k+1) * A396414(k).

Prog

(PARI)
lista(nn, k=3, p=3) = {
  my(T=matrix(nn+1, nn+1, row, col, my(xr=row-1, xc=col-1); if(xc<xr, 0, (xr+1)*binomial(p*xc-(p-1)*xr+1, xc-xr)/(p*xc-(p-1)*xr+1))));
  my(TK=T^k);
  TK[1, ];
};

Crossrefs

Column k=3 of A396412.

Cf. A000108, A001764, A396414.

Sequence in context: A361843 A097188 A025748 * A366085 A394159 A271930

Adjacent sequences: A396412 A396413 A396414 * A396416 A396417 A396418

Keyword

nonn,new

Author

Seiichi Manyama, May 25 2026

Status

approved