A192435 Number of terms in n-th derivative of a function composed with itself n times.

1, 2, 6, 26, 110, 532, 2541, 12644, 63024, 318857, 1618947, 8277062, 42453073, 218597485, 1128527057, 5841301830, 30297014746, 157442596130, 819511659381, 4272054888643, 22299423992018, 116539878029773, 609718298887977, 3193136462042241, 16737951567806110

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..600

W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Formula

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.35270133348664..., c = 0.0504640078963302151598181537452... . - Vaclav Kotesovec, Sep 03 2014, updated May 19 2018

Maple

A:= proc(n, k) option remember;
      `if`(k=1, 1, add(b(n, n, i)*A(i, k-1), i=0..n))
    end:
b:= proc(n, i, k) option remember; `if`(n<k, 0, `if`(n=0, 1, `if`(i<1, 0,
      `if`(n=k, 1, add(b(n-i*j, i-1, k-j), j=0..min(n/i, k))))))
    end:
a:= n-> A(n, n):
seq(a(n), n=1..40);

Mathematica

A[n_, k_] := A[n, k] = If[k == 1, 1, Sum[b[n, n, i]*A[i, k-1], {i, 0, n}]]; b[n_, i_, k_] := b[n, i, k] = If[n<k, 0, If[n == 0, 1, If[i<1, 0, If[n == k, 1,  Sum[b[n - i*j, i-1, k-j], {j, 0, Min[n/i, k]}]]]]]; a[n_] := A[n, n]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

Crossrefs

Main diagonal of A022818.

Cf. A008485.

Sequence in context: A285024 A192403 A282618 * A296217 A050890 A114710

Adjacent sequences: A192432 A192433 A192434 * A192436 A192437 A192438

Keyword

nonn

Author

Alois P. Heinz, Aug 18 2012

Status

approved