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A024208 Number of terms in n-th derivative of a function composed with itself 8 times.

%I A024208 #20 Jul 19 2018 12:16:41
%S A024208 1,1,8,36,148,498,1590,4586,12644,32775,81901,196085,455772,1025779,
%T A024208 2252674,4823546,10116553,20783490,41949270,83211931,162552093,
%U A024208 312850854,594086542,1113610526,2062796698,3777567977,6844786250,12276620372,21809737429,38391720375
%N A024208 Number of terms in n-th derivative of a function composed with itself 8 times.
%D A024208 W. C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.
%H A024208 Alois P. Heinz, <a href="/A024208/b024208.txt">Table of n, a(n) for n = 0..1000</a>
%H A024208 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 A024208 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 A024208 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 A024208 a[n_, k_] := a[n, k] = If[k==1, 1, Sum[b[n, n, i]*a[i, k-1], {i, 0, n}]];
%t A024208 a[n_] := a[n, 8];
%t A024208 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Apr 28 2017, after _Alois P. Heinz_ *)
%Y A024208 Cf. A008778, A022811-A022817, A024207-A024210. First column of A050302.
%Y A024208 Column k=8 of A022818.
%K A024208 nonn
%O A024208 0,3
%A A024208 Winston C. Yang (yang(AT)math.wisc.edu)