A022812 - OEIS

%I #18 Jul 19 2018 12:01:29

%S 1,1,4,10,26,55,121,237,468,867,1597,2821,4952,8421,14206,23439,38324,

%T 61570,98112,154111,240197,370015,565802,856664,1288366,1921016,

%U 2846572,4186730,6122369,8893904,12851713,18460961,26388354,37519159,53101687,74792210

%N Number of terms in n-th derivative of a function composed with itself 4 times.

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

%H Alois P. Heinz, <a href="/A022812/b022812.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, 4]; 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 A039806.

%K nonn

%O 0,3

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