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%I #18 Jul 19 2018 12:11:51
%S 1,1,6,21,71,196,532,1301,3101,6956,15217,31951,65670,130914,256150,
%T 489690,920905,1699693,3092751,5540571,9802091,17114237,29550346,
%U 50444952,85264328,142682505,236649524,389033014,634408230,1026350152,1648328017,2628254619
%N Number of terms in n-th derivative of a function composed with itself 6 times.
%D W. C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.
%H Alois P. Heinz, <a href="/A022814/b022814.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 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 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 a[n_, k_] := a[n, k] = If[k==1, 1, Sum[b[n, n, i]*a[i, k-1], {i, 0, n}]];
%t a[n_] := a[n, 6];
%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Apr 28 2017, after _Alois P. Heinz_ *)
%Y Cf. A008778, A022811-A022818, A024207-A024210. First column of A050300.
%K nonn
%O 0,3
%A Winston C. Yang (yang(AT)math.wisc.edu)
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