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

Author

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

Keywords

• nonn

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.

Crossrefs

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

Column k=10 of A022818.

Programs

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 *)

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).