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

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

Offset: 0

Author

Simon Plouffe

Keywords

• nonn
• easy
• changed

Comments

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

References

• 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.
• 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
• A. G. Shannon, Catalan triangles and Finucan's hidden folders. Notes on Number Theory and Discrete Mathematics, 22(2), 10-16, (2016).
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A181289.

Programs

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

Mathematica

Table[(n+1)*(n+2)*(n+4)*(n+8)*(n+15)/120, {n,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

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

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