A022814 Number of terms in n-th derivative of a function composed with itself 6 times.

1, 1, 6, 21, 71, 196, 532, 1301, 3101, 6956, 15217, 31951, 65670, 130914, 256150, 489690, 920905, 1699693, 3092751, 5540571, 9802091, 17114237, 29550346, 50444952, 85264328, 142682505, 236649524, 389033014, 634408230, 1026350152, 1648328017, 2628254619

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, 6];
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 A050300.

Sequence in context: A169687 A302448 A101904 * A000390 A000391 A360090

Adjacent sequences: A022811 A022812 A022813 * A022815 A022816 A022817

Keyword

nonn

Author

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

Status

approved