A006636 a(n) = (n + 1)*(n + 2)*(n + 4)*(n + 8)*(n + 15)/120. (Formerly M4516)

8, 36, 102, 231, 456, 819, 1372, 2178, 3312, 4862, 6930, 9633, 13104, 17493, 22968, 29716, 37944, 47880, 59774, 73899, 90552, 110055, 132756, 159030, 189280, 223938, 263466, 308357, 359136, 416361, 480624, 552552, 632808, 722092, 821142, 930735, 1051688, 1184859

Offset

0,1

Comments

Former name: From generalized Catalan numbers. - G. C. Greubel, Sep 03 2025

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.

A. G. Shannon, Catalan triangles and Finucan's hidden folders. Notes on Number Theory and Discrete Mathematics, 22(2) (2016), 10-16.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

From Sean A. Irvine, May 30 2017: (Start)

a(n) = (n + 1)*(n + 2)*(n + 4)*(n + 8)*(n + 15)/120.

G.f.: (2 - x)^3/(1 - x)^6. (End)

E.g.f.: exp(x)*(960 + 3360*x + 2280*x^2 + 500*x^3 + 40*x^4 + x^5)/120. - Stefano Spezia, Oct 15 2022

From Amiram Eldar, Sep 29 2025: (Start)

Sum_{n>=0} 1/a(n) = 14530105/84168084.

Sum_{n>=0} (-1)^n/a(n) = 5840*log(2)/7007 - 39860803/84168084. (End)

Mathematica

Table[(n+1)*(n+2)*(n+4)*(n+8)*(n+15)/120, {n, 0, 40}] (* G. C. Greubel, Sep 03 2025 *)

Prog

(Magma)
A006636:= func< n | (n+1)*(n+2)*(n+4)*(n+8)*(n+15)/120 >;
[A006636(n): n in [0..40]]; // G. C. Greubel, Sep 03 2025
(SageMath)
def A006636(n): return (n+1)*(n+2)*(n+4)*(n+8)*(n+15)//120
print([A006636(n) for n in range(41)]) # G. C. Greubel, Sep 03 2025

Crossrefs

Cf. A181289.

Sequence in context: A245360 A032768 A224410 * A325656 A357716 A341987

Adjacent sequences: A006633 A006634 A006635 * A006637 A006638 A006639

Keyword

nonn,easy

Author

Simon Plouffe

Extensions

a(6) and a(8) corrected and more terms from Sean A. Irvine, May 30 2017

New name by G. C. Greubel, Sep 03 2025

Status

approved