A022813 - OEIS

%I #18 Jul 19 2018 12:08:40

%S 1,1,5,15,45,110,271,599,1309,2690,5436,10545,20148,37341,68223,

%T 121878,214846,371993,636570,1073325,1790721,2950922,4816603,7778937,

%U 12455988,19761148,31108121,48572686,75307513,115909727,177255526,269294119,406708721,610593948

%N Number of terms in n-th derivative of a function composed with itself 5 times.

%D W. C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.

%H Alois P. Heinz, <a href="/A022813/b022813.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, 5];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 A039807.

%K nonn

%O 0,3

%A Winston C. Yang (yang(AT)math.wisc.edu)