A104473 a(n) = binomial(n+2,2)*binomial(n+6,2).

15, 63, 168, 360, 675, 1155, 1848, 2808, 4095, 5775, 7920, 10608, 13923, 17955, 22800, 28560, 35343, 43263, 52440, 63000, 75075, 88803, 104328, 121800, 141375, 163215, 187488, 214368, 244035, 276675, 312480, 351648, 394383, 440895, 491400, 546120, 605283, 669123

Offset: 0

Author

Zerinvary Lajos, Apr 18 2005

Keywords

• nonn
• easy

Examples

			a(0) = C(0+2,2)*C(0+6,2) = C(2,2)*C(6,2) = 1*15 = 155.
a(6) = 1*3*5 + 2*4*6 + 3*5*7 + 4*6*8 + 5*7*9 + 6*8*10 + 7*9*11 = 1848.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000217, A000579, A033275, A034856, A062264.

Subsequence of A085780.

Programs

Magma

[Binomial(n+2, 2)*Binomial(n+6, 2): n in [0..50]]; // Vincenzo Librandi, Apr 28 2014

Mathematica

f[n_] := Binomial[n + 2, 2] Binomial[n + 6, 2]; Table[f[n], {n,0,40}] (* Robert G. Wilson v, Apr 20 2005 *)
CoefficientList[Series[3 (5-4*x+x^2)/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Apr 28 2014 *)

PARI

a(n)=binomial(n+2,2)*binomial(n+6,2) \\ Charles R Greathouse IV, Jun 07 2013

SageMath

def A104473(n): return binomial(n+2,2)*binomial(n+6,2)
print([A104473(n) for n in range(51)]) # G. C. Greubel, Mar 05 2025

Formula

a(n) = (1/4)*(n+1)*(n+2)*(n+5)*(n+6).

a(n) = A034856(n+2)^2 - 1. - J. M. Bergot, Dec 14 2010

G.f.: 3*(5-4*x+x^2)/(1-x)^5. - Colin Barker, Sep 21 2012

a(n) = Sum_{i=1..n+1} i*(i+2)*(i+4). - Bruno Berselli, Apr 28 2014

a(n) = A000217(n)*A000217(n+4) = 3*A033275(n+4). - R. J. Mathar, Nov 29 2015

From Amiram Eldar, Aug 30 2022: (Start)

Sum_{n>=0} 1/a(n) = 43/450.

Sum_{n>=0} (-1)^n/a(n) = 16*log(2)/15 - 154/225. (End)

From G. C. Greubel, Mar 05 2025: (Start)

a(n) = 90*A000579(n+6)/A000279(n+3).

E.g.f.: (1/4)*(60 + 192*x + 114*x^2 + 20*x^3 + x^4)*exp(x). (End)