A105251 a(n) = binomial(n+4,n)*binomial(n+8,n).

1, 45, 675, 5775, 34650, 162162, 630630, 2123550, 6370650, 17381650, 43801758, 103169430, 229265400, 484306200, 978496200, 1900457064, 3563356995, 6473226375, 11428041625, 19658764125, 33026723730, 54295634250, 87501228750, 138447123750, 215362192500

Offset: 0

Author

Zerinvary Lajos, Apr 14 2005

Keywords

• easy
• nonn

Examples

			a(0): C(0+4,0)*C(0+8,0) = C(4,0)*C(8,0) = 1*1 = 1;
a(10): C(10+4,10)*C(10+8,10) = C(14,10)*(18,10) = 1001*43758 = 43801758.
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Crossrefs

Cf. A062264.

Programs

Magma

[Binomial(n+4,n)*Binomial(n+8,n): n in [0..30]]; // Vincenzo Librandi, Jul 31 2015

Mathematica

f[n_] := Binomial[n + 4, n]Binomial[n + 8, n]; Table[ f[n], {n, 0, 22}] (* Robert G. Wilson v, Apr 20 2005 *)
LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{1,45,675,5775,34650,162162,630630,2123550,6370650,17381650,43801758,103169430,229265400},30] (* Harvey P. Dale, May 19 2024 *)

SageMath

def A105251(n): return binomial(n+4,n)*binomial(n+8,n)
print([A105251(n) for n in range(31)]) # G. C. Greubel, Mar 04 2025

Formula

G.f.: (1+32*x+168*x^2+224*x^3+70*x^4)/(1-x)^13. - Colin Barker, Jan 21 2013

From Amiram Eldar, Sep 01 2022: (Start)

Sum_{n>=0} 1/a(n) = 640*Pi^2 - 13925728/2205.

Sum_{n>=0} (-1)^n/a(n) = 640*Pi^2/3 - 90112*log(2)/105 + 471984/1225. (End)

Extensions

More terms from Robert G. Wilson v, Apr 20 2005