A104675 a(n) = C(n+1,n) * C(n+6,1).

6, 14, 24, 36, 50, 66, 84, 104, 126, 150, 176, 204, 234, 266, 300, 336, 374, 414, 456, 500, 546, 594, 644, 696, 750, 806, 864, 924, 986, 1050, 1116, 1184, 1254, 1326, 1400, 1476, 1554, 1634, 1716, 1800, 1886, 1974, 2064, 2156, 2250, 2346, 2444, 2544, 2646

Offset: 0

Author

Zerinvary Lajos, Apr 22 2005

Keywords

• easy
• nonn

Examples

			If n=0 then C(0+1,0+0) * C(0+6,1) = C(1,0) * C(6,1) = 1*6 = 6.
If n=5 then C(5+1,5+0) * C(5+6,1) = C(6,5) * C(11,1) = 6*11 = 66.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A002522, A028557, A062190.

Programs

Magma

[(n+1)*(n+6): n in [0..50]]; // G. C. Greubel, Mar 01 2025

Mathematica

Table[Binomial[n + 1, n] Binomial[n + 6, 1], {n, 0, 48}] (* or *)
CoefficientList[Series[2 (3 - 2 x)/(1 - x)^3, {x, 0, 49}], x] (* or *)
LinearRecurrence[{3, -3, 1}, {6, 14, 24}, 49] (* Michael De Vlieger, Apr 06 2017 *)

PARI

Vec(2*(3 - 2*x) / (1 - x)^3 + O(x^80)) \\ Colin Barker, Apr 06 2017

PARI

a(n)=(n+6)*(n+1) \\ Charles R Greathouse IV, Jun 17 2017

Python

from sympy import binomial
def a(n): return binomial(n + 1, n) * binomial(n + 6, 1) # Indranil Ghosh, Apr 06 2017

Formula

a(n) = (n+1)*(n+6) = A028557(n+1). - R. J. Mathar, May 19 2008

a(n) = 2*n + a(n-1) + 6 (with a(0)=6). Vincenzo Librandi, Nov 13 2010

From Colin Barker, Apr 06 2017: (Start)

G.f.: 2*(3 - 2*x) / (1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. (End)

E.g.f.: exp(x)*(x^2 + 8x + 6). - Indranil Ghosh, Apr 06 2017

From Amiram Eldar, Aug 30 2022: (Start)

Sum_{n>=0} 1/a(n) = 137/300.

Sum_{n>=0} (-1)^n/a(n) = 2*log(2)/5 - 47/300. (End)