A022815 Number of terms in 5th derivative of a function composed with itself n times.

1, 7, 23, 55, 110, 196, 322, 498, 735, 1045, 1441, 1937, 2548, 3290, 4180, 5236, 6477, 7923, 9595, 11515, 13706, 16192, 18998, 22150, 25675, 29601, 33957, 38773, 44080, 49910, 56296, 63272, 70873, 79135, 88095, 97791, 108262, 119548, 131690, 144730, 158711, 173677

Offset

1,2

References

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = n*(n+1)*(n^2+13*n-2)/24. - John W. Layman, Apr 27 2000

G.f.: x*(1-2*x^2+2*x)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; corrected by R. J. Mathar, Sep 16 2009

a(n) = n*A000217(n) + sum((n-i)*A000217(i), i=0..n-1). - Bruno Berselli, Jun 23 2013

E.g.f.: (1/24)*x*(24 + 60*x + 20*x^2 + x^3)*exp(x). - G. C. Greubel, Dec 26 2025

Example

a(7) = 7*28 + (7*0+6*1+5*3+4*6+3*10+2*15+1*21) = 322. [Bruno Berselli, Jun 22 2013]

Mathematica

Table[n*(n+1)*(n^2+13*n-2)/24, {n, 50}] (* G. C. Greubel, Dec 26 2025 *)

Prog

(Magma) [n*(n+1)*(n^2+13*n-2)/24: n in [1..40]]; // Vincenzo Librandi, Oct 10 2011
(SageMath)
def A022815(n): return n*(n+1)*(n**2+13*n-2)//24 # G. C. Greubel, Dec 26 2025

Crossrefs

Cf. A000217, A008778, A022811-A022818, A024207-A024210.

Sequence in context: A211791 A004068 A261893 * A172252 A027116 A151718

Adjacent sequences: A022812 A022813 A022814 * A022816 A022817 A022818

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Christian G. Bower, Aug 15 1999

Status

approved