A022813 Number of terms in n-th derivative of a function composed with itself 5 times.

1, 1, 5, 15, 45, 110, 271, 599, 1309, 2690, 5436, 10545, 20148, 37341, 68223, 121878, 214846, 371993, 636570, 1073325, 1790721, 2950922, 4816603, 7778937, 12455988, 19761148, 31108121, 48572686, 75307513, 115909727, 177255526, 269294119, 406708721, 610593948

Offset

0,3

References

W. C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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

Formula

If a(n,m) = number of terms in m-derivative of a function composed with itself n times, p(n,k) = number of partitions of n into k parts, then a(n,m) = sum_{i=0..m} p(m,i)*a(n-1,i).

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n<k, 0, If[n==0, 1, If[i<1, 0, Sum[b[n-i*j, i-1, k-j], {j, 0, Min[n/i, k]}]]]];
a[n_, k_] := a[n, k] = If[k==1, 1, Sum[b[n, n, i]*a[i, k-1], {i, 0, n}]];
a[n_]:=a[n, 5]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 28 2017, after Alois P. Heinz *)

Crossrefs

Cf. A008778, A022811-A022818, A024207-A024210. First column of A039807.

Sequence in context: A094283 A373300 A158875 * A000334 A000335 A271180

Adjacent sequences: A022810 A022811 A022812 * A022814 A022815 A022816

Keyword

nonn

Author

Winston C. Yang (yang(AT)math.wisc.edu)

Status

approved