A064059 Seventh column of Catalan triangle A009766.

132, 429, 1001, 2002, 3640, 6188, 9996, 15504, 23256, 33915, 48279, 67298, 92092, 123970, 164450, 215280, 278460, 356265, 451269, 566370, 704816, 870232, 1066648, 1298528, 1570800, 1888887, 2258739, 2686866, 3180372, 3746990, 4395118, 5133856, 5973044

Offset: 0

Author

Wolfdieter Lang, Sep 13 2001

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000096, A005586, A005587, A005557 (third to sixth column).

Programs

Magma

A064059:= func< n | (n+1)*Binomial(n+12,5)/6 >;
[A064059(n): n in [0..40]]; // G. C. Greubel, Sep 27 2024

Maple

[seq(binomial(n+1,6)-2*binomial(n,5),n=12..55)]; # Zerinvary Lajos, Jul 19 2006

Mathematica

CoefficientList[Series[(42 z^5-252 z^4+616 z^3-770 z^2+495 z-132)/(z-1)^7, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{132,429,1001,2002,3640,6188,9996},40] (* Harvey P. Dale, Jan 08 2025 *)

SageMath

def A064059(n): return (n+1)*binomial(n+12,5)//6
[A064059(n) for n in range(41)] # G. C. Greubel, Sep 27 2024

Formula

G.f.: (132-495*x+770*x^2-616*x^3+252*x^4-42*x^5)/(1-x)^7; numerator polynomial is N(2;5, x) from A062991.

a(n) = A009766(n+6, 6) = (n+1)*binomial(n+12,5)/6.

a(n) = binomial(n+13,6) - 2*binomial(n+12,5). - Zerinvary Lajos, Jul 19 2006

a(n) = A214292(n+11,5). - Reinhard Zumkeller, Jul 12 2012

From Amiram Eldar, Sep 06 2022: (Start)

Sum_{n>=0} 1/a(n) = 25961/2134440.

Sum_{n>=0} (-1)^n/a(n) = 4160*log(2)/77 - 79917773/2134440. (End)