A396592 Expansion of e.g.f. F_2(x)/x and F_k(x) is the k-th iteration of x*G(x) with G(x) = exp(x*G(x)^4).

1, 2, 22, 479, 15880, 711977, 40372828, 2773990837, 224184734080, 20850884618561, 2194931003624164, 258095100967003445, 33540797497557829360, 4775042199876691978081, 739225290389472515211076, 123660840486107821484582093, 22232061279653818569516521344, 4275230842901062415074330238081

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

E.g.f.: (1/x) * Series_Reversion( H_2(x) ), where H_k(x) is the k-th iterate of U(x)*exp(-4*U(x)) and U(x) = -LambertW(-3*x)/3.

a(n) = Sum_{k=0..n} (k+1) * (4*k+1)^(k-1) * (4*n-3*k+1)^(n-k-1) * binomial(n,k).

Mathematica

a[n_]:=Sum[(k+1) * (4*k+1)^(k-1) * (4*n-3*k+1)^(n-k-1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* Stefano Spezia, May 30 2026 *)

Prog

(PARI)
lista(nn, k=2, p=4, s=1, r=1) = {
  my(T=matrix(nn+1, nn+1, row, col, my(xr=row-1, xc=col-1); if(xc<xr, 0, (s*xr+r)*(p*xc-(p-s)*xr+r)^(xc-xr-1)*binomial(xc, xr))));
  my(TK=T^k);
  TK[1, ];
};

Crossrefs

Column k=2 of A396588.

Cf. A101878, A396589, A396590.

Sequence in context: A084949 A276454 A377633 * A137076 A090730 A090313

Adjacent sequences: A396588 A396589 A396590 * A396593 A396594 A396597

Keyword

nonn

Author

Seiichi Manyama, May 29 2026

Status

approved