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A024210 Number of terms in n-th derivative of a function composed with itself 10 times. 14
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A022575 A202481 A169720 * A253002 A316109 A188168

Adjacent sequences:  A024207 A024208 A024209 * A024211 A024212 A024213

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified October 27 13:45 EDT 2021. Contains 348276 sequences. (Running on oeis4.)