A022812 Number of terms in n-th derivative of a function composed with itself 4 times.

1, 1, 4, 10, 26, 55, 121, 237, 468, 867, 1597, 2821, 4952, 8421, 14206, 23439, 38324, 61570, 98112, 154111, 240197, 370015, 565802, 856664, 1288366, 1921016, 2846572, 4186730, 6122369, 8893904, 12851713, 18460961, 26388354, 37519159, 53101687, 74792210

Offset: 0

Author

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

Keywords

• nonn

References

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Crossrefs

Cf. A008778, A022811-A022818, A024207-A024210. First column of A039806.

Programs

Mathematica

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]}]]]];
a[n_, k_] := a[n, k] = If[k == 1, 1, Sum[b[n, n, i]*a[i, k-1], {i, 0, n}]];
a[n_] := a[n, 4]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 28 2017, after Alois P. Heinz *)

Formula

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).