A107422 a(n) = binomial(n+7,7) * binomial(n+10,10).

1, 88, 2376, 34320, 330330, 2378376, 13741728, 66745536, 281582730, 1056804320, 3593134688, 11224833984, 32583198648, 88687996320, 228054847680, 557467405440, 1302209017395, 2919831983640, 6308278977000, 13175740578000, 26680874670450, 52514737446600

Offset: 0

Author

Zerinvary Lajos, May 26 2005

Keywords

• easy
• nonn

Examples

			If n=0 then C(0+7,7)*C(0+10,10) = C(7,7)*C(10,10) = 1*1 = 1.
If n=4 then C(4+7,7)*C(4+10,10) = C(11,7)*C(14,10) = 330*1001 = 330330.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (18,-153,816,-3060,8568,-18564,31824,-43758,48620,-43758,31824,-18564,8568,-3060,816,-153,18,-1).

Crossrefs

Cf. A062145.

Programs

Magma

A107422:= func< n | Binomial(n+7,n)*Binomial(n+10,n) >;
[A107422(n): n in [0..40]]; // G. C. Greubel, Mar 09 2025

Mathematica

a[n_] := Binomial[n + 7, 7] * Binomial[n + 10, 10]; Array[a, 25, 0] (* Amiram Eldar, Sep 01 2022 *)

PARI

a(n)={binomial(n+7,7) * binomial(n+10,10)} \\ Andrew Howroyd, Nov 08 2019

SageMath

def A107422(n): return binomial(n+7,n)*binomial(n+10,n)
print([A107422(n) for n in range(41)]) # G. C. Greubel, Mar 09 2025

Formula

From Chai Wah Wu, Apr 10 2021: (Start)

a(n) = 18*a(n-1) - 153*a(n-2) + 816*a(n-3) - 3060*a(n-4) + 8568*a(n-5) - 18564*a(n-6) + 31824*a(n-7) - 43758*a(n-8) + 48620*a(n-9) - 43758*a(n-10) + 31824*a(n-11) - 18564*a(n-12) + 8568*a(n-13) - 3060*a(n-14) + 816*a(n-15) - 153*a(n-16) + 18*a(n-17) - a(n-18) for n > 17.

G.f.: (1 + 70*x + 945*x^2 + 4200*x^3 + 7350*x^4 + 5292*x^5 + 1470*x^6 + 120*x^7)/(1 - x)^18. (End)

From Amiram Eldar, Sep 01 2022: (Start)

Sum_{n>=0} 1/a(n) = 175175*Pi^2/3 - 1493773847/2592.

Sum_{n>=0} (-1)^n/a(n) = 875*Pi^2/6 + 90112*log(2)/9 - 760090799/90720. (End)

Extensions

Terms a(8) and beyond from Andrew Howroyd, Nov 08 2019