A107419 a(n) = binomial(n+4,4)*binomial(n+7,7).

1, 40, 540, 4200, 23100, 99792, 360360, 1132560, 3185325, 8179600, 19467448, 43439760, 91706160, 184497600, 355816800, 661028544, 1187785665, 2071432440, 3516320500, 5824819000, 9436206780, 14978106000, 23333661000, 35728290000, 53840548125, 79942445856

Offset: 0

Author

Zerinvary Lajos, May 26 2005

Keywords

• easy
• nonn

Examples

			a(0) = C(0+4,4)*C(0+7,7) = C(4,4)*C(7,7) = 1*1 = 1.
a(6) = C(6+4,4)*C(6+7,7) = C(10,4)*C(13,7) = 210*1716 = 360360.
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Crossrefs

Cf. A062145.

Programs

Magma

A107419:= func< n | Binomial(n+4,n)*Binomial(n+7,n) >;
[A107419(n): n in [0..40]]; // G. C. Greubel, Mar 10 2025

Mathematica

Table[Binomial[n+4,4]Binomial[n+7,7],{n,0,30}] (* Harvey P. Dale, Jul 27 2019 *)

PARI

for(n=0,29,print1(binomial(n+4,4)*binomial(n+7,7),","))

SageMath

def A107419(n): return binomial(n+4,n)*binomial(n+7,n)
print([A107419(n) for n in range(41)]) # G. C. Greubel, Mar 10 2025

Formula

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

a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12) for n > 11.

G.f.: (1 + 28*x + 126*x^2 + 140*x^3 + 35*x^4)/(1 - x)^12. (End)

From Amiram Eldar, Sep 06 2022: (Start)

Sum_{n>=0} 1/a(n) = 392*Pi^2 - 870268/225.

Sum_{n>=0} (-1)^n/a(n) = 56*Pi^2/3 - 3584*log(2)/15 - 441/25. (End)

Extensions

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