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

%I A006636 M4516 #33 Sep 03 2025 07:39:02
%S A006636 8,36,102,231,456,819,1372,2178,3312,4862,6930,9633,13104,17493,22968,
%T A006636 29716,37944,47880,59774,73899,90552,110055,132756,159030,189280,
%U A006636 223938,263466,308357,359136,416361,480624,552552,632808,722092,821142,930735
%N A006636 a(n) = (n + 1)*(n + 2)*(n + 4)*(n + 8)*(n + 15)/120.
%C A006636 Former name: From generalized Catalan numbers. - _G. C. Greubel_, Sep 03 2025
%D A006636 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.
%D A006636 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006636 G. C. Greubel, <a href="/A006636/b006636.txt">Table of n, a(n) for n = 0..1000</a>
%H A006636 A. G. Shannon, <a href="https://nntdm.net/volume-22-2016/number-2/10-16/">Catalan triangles and Finucan's hidden folders</a>. Notes on Number Theory and Discrete Mathematics, 22(2), 10-16, (2016).
%H A006636 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A006636 From _Sean A. Irvine_, May 30 2017: (Start)
%F A006636 a(n) = (n + 1)*(n + 2)*(n + 4)*(n + 8)*(n + 15)/120.
%F A006636 G.f.: (2 - x)^3/(1 - x)^6. (End)
%F A006636 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
%t A006636 Table[(n+1)*(n+2)*(n+4)*(n+8)*(n+15)/120, {n,0,40}] (* _G. C. Greubel_, Sep 03 2025 *)
%o A006636 (Magma)
%o A006636 A006636:= func< n | (n+1)*(n+2)*(n+4)*(n+8)*(n+15)/120 >;
%o A006636 [A006636(n): n in [0..40]]; // _G. C. Greubel_, Sep 03 2025
%o A006636 (SageMath)
%o A006636 def A006636(n): return (n+1)*(n+2)*(n+4)*(n+8)*(n+15)//120
%o A006636 print([A006636(n) for n in range(41)]) # _G. C. Greubel_, Sep 03 2025
%Y A006636 Cf. A181289.
%K A006636 nonn,easy,changed
%O A006636 0,1
%A A006636 _Simon Plouffe_
%E A006636 a(6) and a(8) corrected and more terms from _Sean A. Irvine_, May 30 2017
%E A006636 New name by _G. C. Greubel_, Sep 03 2025