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A192435 Number of terms in n-th derivative of a function composed with itself n times.

%I A192435 #37 Jul 19 2018 16:34:47
%S A192435 1,2,6,26,110,532,2541,12644,63024,318857,1618947,8277062,42453073,
%T A192435 218597485,1128527057,5841301830,30297014746,157442596130,
%U A192435 819511659381,4272054888643,22299423992018,116539878029773,609718298887977,3193136462042241,16737951567806110
%N A192435 Number of terms in n-th derivative of a function composed with itself n times.
%H A192435 Alois P. Heinz, <a href="/A192435/b192435.txt">Table of n, a(n) for n = 1..600</a>
%H A192435 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 A192435 a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.35270133348664..., c = 0.0504640078963302151598181537452... . - _Vaclav Kotesovec_, Sep 03 2014, updated May 19 2018
%p A192435 A:= proc(n, k) option remember;
%p A192435       `if`(k=1, 1, add(b(n, n, i)*A(i, k-1), i=0..n))
%p A192435     end:
%p A192435 b:= proc(n, i, k) option remember; `if`(n<k, 0, `if`(n=0, 1, `if`(i<1, 0,
%p A192435       `if`(n=k, 1, add(b(n-i*j, i-1, k-j), j=0..min(n/i, k))))))
%p A192435     end:
%p A192435 a:= n-> A(n, n):
%p A192435 seq(a(n), n=1..40);
%t A192435 A[n_, k_] := A[n, k] = If[k == 1, 1, Sum[b[n, n, i]*A[i, k-1], {i, 0, n}]]; b[n_, i_, k_] := b[n, i, k] = If[n<k, 0, If[n == 0, 1, If[i<1, 0, If[n == k, 1,  Sum[b[n - i*j, i-1, k-j], {j, 0, Min[n/i, k]}]]]]]; a[n_] := A[n, n]; Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Feb 05 2015, after _Alois P. Heinz_ *)
%Y A192435 Main diagonal of A022818.
%Y A192435 Cf. A008485.
%K A192435 nonn
%O A192435 1,2
%A A192435 _Alois P. Heinz_, Aug 18 2012