A024208 Number of terms in n-th derivative of a function composed with itself 8 times.

1, 1, 8, 36, 148, 498, 1590, 4586, 12644, 32775, 81901, 196085, 455772, 1025779, 2252674, 4823546, 10116553, 20783490, 41949270, 83211931, 162552093, 312850854, 594086542, 1113610526, 2062796698, 3777567977, 6844786250, 12276620372, 21809737429, 38391720375

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, 8];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 28 2017, after Alois P. Heinz *)

Crossrefs

Cf. A008778, A022811-A022817, A024207-A024210. First column of A050302.

Column k=8 of A022818.

Sequence in context: A121255 A210656 A119767 * A000427 A000428 A083597

Adjacent sequences: A024205 A024206 A024207 * A024209 A024210 A024211

Keyword

nonn

Author

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

Status

approved