A024210 Number of terms in n-th derivative of a function composed with itself 10 times.

1, 1, 10, 55, 265, 1045, 3817, 12583, 39148, 114235, 318857, 850576, 2190850, 5451721, 13184711, 31023842, 71286349, 160139911, 352574213, 761567304, 1616713932, 3376143283, 6944345483, 14080091227, 28169087367, 55644767253, 108617341172, 209626751905

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

Crossrefs

Cf. A008778, A022811-A022817, A024207-A024209. First column of A050304.

Column k=10 of A022818.

Sequence in context: A202481 A348663 A169720 * A253002 A357668 A316109

Adjacent sequences: A024207 A024208 A024209 * A024211 A024212 A024213

Keyword

nonn

Author

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

Status

approved