A395269 E.g.f. A(x) satisfies A'(x) = 1 + A^5(x), where A^l(x) denotes the l-th iterate of A, with A(0) = 0.

1, 1, 5, 55, 985, 25075, 840900, 35394960, 1808623810, 109496294445, 7710132048400, 622342364799875, 56914610070416925, 5840747950246941200, 667216684038723886425, 84264849101138358291100, 11696222862477631614890825, 1775132042879489872026289950, 293246294566669557714559409325

Offset

1,3

Links

Seiichi Manyama, Table of n, a(n) for n = 1..286

Formula

E.g.f. A(x) satisfies A'(x) = 1 + A^l(x), where A^l(x) denotes the l-th iterate of A, with A(0) = 0.

A(x) = Series_Reversion( Integral 1/(1 + A^{l-1}(x)) dx ). for l > 0.

Let a(n,k,l) = n! * [x^n] A^k(x), where A^k(x) is the k-th iterate of A.

a(n,0,l) = 0^(n-1) and a(n,k,l) = a(n,k-1,l) + Sum_{j=1..n-1} binomial(n-1,j) * a(j,k+l-1,l) * a(n-j,k-1,l) for k > 0.

Prog

(PARI)
a_vector(n, k=1, l=5) = {
  my(k_limit(row)=k+(n-row)*l, A=vector(n, row, vector(k_limit(row)+1)));
  for(col=0, k_limit(1), A[1][col+1]=1);
  for(row=2, n, A[row][1]=0);
  for(row=2, n,
    for(col=1, k_limit(row),
      A[row][col+1]=A[row][col]+sum(j=1, row-1, binomial(row-1, j)*A[j][col+l]*A[row-j][col]);
    );
  );
  vector(n, row, A[row][k+1])
};

Crossrefs

Column k=1 of A396973.

Cf. A001028, A193098, A193099.

Sequence in context: A155807 A135861 A141361 * A203013 A266481 A371316

Adjacent sequences: A395266 A395267 A395268 * A395270 A395271 A395272

Keyword

nonn

Author

Seiichi Manyama, Jun 11 2026

Status

approved