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A024209 Number of terms in n-th derivative of a function composed with itself 9 times. 5
1, 1, 9, 45, 201, 735, 2517, 7785, 22857, 63024, 166819, 422537, 1035971, 2456694, 5672347, 12756334, 28053280, 60371967, 127479247, 264311585, 539102751, 1082474167, 2142579168, 4183251750, 8064722973, 15360809911, 28928858208, 53896616704, 99398216733 (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, 9];

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

CROSSREFS

Cf. A008778, A022811-A022817, A024207-A024210. First column of A050303.

Column k=9 of A022818.

Sequence in context: A050574 A124647 A111640 * A179855 A255965 A180796

Adjacent sequences:  A024206 A024207 A024208 * A024210 A024211 A024212

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified April 20 22:39 EDT 2021. Contains 343140 sequences. (Running on oeis4.)