A107418 a(n) = binomial(n+3,3)*binomial(n+6,6).

1, 28, 280, 1680, 7350, 25872, 77616, 205920, 495495, 1101100, 2290288, 4504864, 8446620, 15193920, 26356800, 44279424, 72299997, 115079580, 179012680, 272734000, 407737330, 599124240, 866502000, 1235052000, 1736791875, 2412056556, 3311225568, 4496726080, 6045343480

Offset: 0

Author

Zerinvary Lajos, May 26 2005

Keywords

• easy
• nonn

Examples

			If n=0 then C(0+3,3)*C(0+6,6) = C(3,3)*C(6,6) = 1*1 = 1.
If n=8 then C(8+3,3)*C(8+6,6) = C(11,3)*C(14,6) = 165*3003 = 495495.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Crossrefs

Cf. A062145.

Programs

Magma

A107418:= func< n | Binomial(n+3,n)*Binomial(n+6,n) >;
[A107418(n): n in [0..40]]; // G. C. Greubel, Mar 10 2025

Maple

seq(binomial(n+3,3)*binomial(n+6,6),n=0..100); # Robert Israel, Feb 24 2017

Mathematica

a[n_] := Binomial[n + 3, 3] * Binomial[n + 6, 6]; Array[a, 30, 0] (* Amiram Eldar, Sep 06 2022 *)

PARI

for(n=0,29,print1(binomial(n+3,3)*binomial(n+6,6),","))

SageMath

def A107418(n): return binomial(n+3,n)*binomial(n+6,n)
print([A107418(n) for n in range(41)]) # G. C. Greubel, Mar 10 2025

Formula

G.f.: (1 + 18*x + 45*x^2 + 20*x^3)/(1-x)^10. - Robert Israel, Feb 24 2017

From Amiram Eldar, Sep 06 2022: (Start)

Sum_{n>=0} 1/a(n) = 63*Pi^2 - 124149/200.

Sum_{n>=0} (-1)^n/a(n) = 3*Pi^2/2 + 1344*log(2)/5 - 40031/200. (End)

Extensions

Corrected and extended by Rick L. Shepherd, May 27 2005