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A024207 Number of terms in n-th derivative of a function composed with itself 7 times.

%I A024207 #20 Jul 19 2018 12:14:56
%S A024207 1,1,7,28,105,322,952,2541,6539,15833,37148,83594,183289,389520,
%T A024207 809820,1643375,3272797,6390745,12279337,23208483,43252360,79483096,
%U A024207 144265338,258673983,458747540,804877837,1398356706,2406328974,4104352128,6940717598,11643270856
%N A024207 Number of terms in n-th derivative of a function composed with itself 7 times.
%D A024207 W. C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.
%H A024207 Alois P. Heinz, <a href="/A024207/b024207.txt">Table of n, a(n) for n = 0..1000</a>
%H A024207 W. C. Yang, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00412-4">Derivatives are essentially integer partitions</a>, Discrete Mathematics, 222(1-3), July 2000, 235-245.
%F A024207 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).
%t A024207 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]}]]]];
%t A024207 a[n_, k_] := a[n, k] = If[k == 1, 1, Sum[b[n, n, i]*a[i, k-1], {i, 0, n}]];
%t A024207 a[n_] := a[n, 7];
%t A024207 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Apr 28 2017, after _Alois P. Heinz_ *)
%Y A024207 Cf. A008778, A022811-A022817, A024208-A024210. First column of A050301.
%Y A024207 Column k=7 of A022818.
%K A024207 nonn
%O A024207 0,3
%A A024207 Winston C. Yang (yang(AT)math.wisc.edu)
%E A024207 More terms from _Alois P. Heinz_, Aug 18 2012