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A024210 Number of terms in n-th derivative of a function composed with itself 10 times.

%I A024210 #20 Jul 19 2018 12:20:49
%S A024210 1,1,10,55,265,1045,3817,12583,39148,114235,318857,850576,2190850,
%T A024210 5451721,13184711,31023842,71286349,160139911,352574213,761567304,
%U A024210 1616713932,3376143283,6944345483,14080091227,28169087367,55644767253,108617341172,209626751905
%N A024210 Number of terms in n-th derivative of a function composed with itself 10 times.
%D A024210 W. C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.
%H A024210 Alois P. Heinz, <a href="/A024210/b024210.txt">Table of n, a(n) for n = 0..1000</a>
%H A024210 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 A024210 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 A024210 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 A024210 a[n_, k_] := a[n, k] = If[k == 1, 1, Sum[b[n, n, i]*a[i, k-1], {i, 0, n}]];
%t A024210 a[n_] := a[n, 10]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Apr 28 2017, after _Alois P. Heinz_ *)
%Y A024210 Cf. A008778, A022811-A022817, A024207-A024209. First column of A050304.
%Y A024210 Column k=10 of A022818.
%K A024210 nonn
%O A024210 0,3
%A A024210 Winston C. Yang (yang(AT)math.wisc.edu)