A024207 Number of terms in n-th derivative of a function composed with itself 7 times.
1, 1, 7, 28, 105, 322, 952, 2541, 6539, 15833, 37148, 83594, 183289, 389520, 809820, 1643375, 3272797, 6390745, 12279337, 23208483, 43252360, 79483096, 144265338, 258673983, 458747540, 804877837, 1398356706, 2406328974, 4104352128, 6940717598, 11643270856
Offset: 0
Keywords
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
Programs
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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, 7]; 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).
Extensions
More terms from Alois P. Heinz, Aug 18 2012