A105249 a(n) = binomial(n+2,n)*binomial(n+6,n).

1, 21, 168, 840, 3150, 9702, 25872, 61776, 135135, 275275, 528528, 965328, 1689324, 2848860, 4651200, 7379904, 11415789, 17261937, 25573240, 37191000, 53183130, 74890530, 103980240, 142506000, 192976875, 258434631, 342540576, 449672608, 585033240, 754769400

Offset: 0

Author

Zerinvary Lajos, Apr 14 2005

Keywords

• easy
• nonn

Examples

			a(0): C(0+2,0)*C(0+6,0) = C(2,0)*C(6,0) = 1*1 = 1;
a(10): C(10+2,10)*C(10+6,10) = C(12,10)*C(16,10) = 66*8008 = 528528.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A062264.

Programs

Magma

[Binomial(n+2,n)*Binomial(n+6,n): n in [0..30]]; // Vincenzo Librandi, Jul 31 2015

Mathematica

f[n_] := Binomial[n + 2, n]Binomial[n + 6, n]; Table[ f[n], {n, 0, 27}] (* Robert G. Wilson v, Apr 20 2005 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,21,168,840,3150,9702,25872,61776,135135},30] (* Harvey P. Dale, Oct 08 2012 *)

SageMath

def A105249(n): return binomial(n+2,n)*binomial(n+6,n)
print([a105249(n) for n in range(31)]) # G. C. Greubel, Mar 04 2025

Formula

a(0)=1, a(1)=21, a(2)=168, a(3)=840, a(4)=3150, a(5)=9702, a(6)=25872, a(7)=61776, a(8)=135135, a(n) = 9*a(n-1) -36*a(n-2) +84*a(n-3) -126*a(n-4) +126*a(n-5) -84*a(n-6) +36*a(n-7) -9*a(n-8) +a(n-9). - Harvey P. Dale, Oct 08 2012

G.f.: (1+12*x+15*x^2)/(1-x)^9. - Colin Barker, Jan 21 2013

From Amiram Eldar, Sep 04 2022: (Start)

Sum_{n>=0} 1/a(n) = 12*Pi^2 - 5869/50.

Sum_{n>=0} (-1)^n/a(n) = 256*log(2)/5 - 4*Pi^2 + 371/75. (End)

E.g.f.: (1/1440)*(1440 + 28800*x + 91440*x^2 + 95520*x^3 + 42900*x^4 + 9312*x^5 + 1010*x^6 + 52*x^7 + x^8)*exp(x). - G. C. Greubel, Mar 04 2025

Extensions

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