A064061 Eighth column of Catalan triangle A009766.

429, 1430, 3432, 7072, 13260, 23256, 38760, 62016, 95931, 144210, 211508, 303600, 427570, 592020, 807300, 1085760, 1442025, 1893294, 2459664, 3164480, 4034712, 5101360, 6399888, 7970688, 9859575, 12118314, 14805180, 17985552, 21732542, 26127660, 31261516

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 (8,-28,56,-70,56,-28,8,-1).

Crossrefs

Cf. A009766, A064059 (seventh column), A062991, A214292.

Programs

Magma

A064061:= func< n | (n+1)*Binomial(n+14, 6)/7 >;
[A064061(n): n in [0..40]]; // G. C. Greubel, Sep 28 2024

Maple

[seq(binomial(n,7)-binomial(n,5),n=13..37)]; # Zerinvary Lajos, Nov 25 2006

Mathematica

CoefficientList[Series[(132*z^6 - 924*z^5 + 2730*z^4 - 4368*z^3 + 4004*z^2 - 2002*z + 429)/(z - 1)^8, {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 16 2011 *)
Table[Binomial[n,7]-Binomial[n,5],{n,13,50}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{429,1430,3432,7072,13260,23256,38760,62016},40] (* Harvey P. Dale, Sep 03 2015 *)

SageMath

def A064061(n): return (n+1)*binomial(n+14,6)//7
[A064061(n) for n in range(41)] # G. C. Greubel, Sep 28 2024

Formula

a(n) = A009766(n+7, 7) = (n+1)*binomial(n+14, 6)/7.

G.f.: (429-2002*x+4004*x^2-4368*x^3+2730* x^4-924*x^5+132*x^6)/(1-x)^8; numerator polynomial is N(2;6, x) from A062991.

a(n) = C(n+13,7) - C(n+13,5). - Zerinvary Lajos, Nov 25 2006

a(n) = A214292(n+13,6). - Reinhard Zumkeller, Jul 12 2012

From Amiram Eldar, Sep 06 2022: (Start)

Sum_{n>=0} 1/a(n) = 323171/88339680.

Sum_{n>=0} (-1)^n/a(n) = 7929257917/88339680 - 55552*log(2)/429. (End)