A105942 a(n) = binomial(n+6,6)*binomial(n+9,6).

84, 1470, 12936, 77616, 360360, 1387386, 4624620, 13741728, 37165128, 92912820, 217273056, 479693760, 1007356896, 2024399916, 3912705720, 7303717344, 13213962300, 23241027810, 39841761960, 66720654000, 109363854600, 175763337750, 277386503940, 430459323840

Offset: 0

Author

Zerinvary Lajos, Apr 27 2005

Keywords

• easy
• nonn

Examples

			If n=0 then C(0+6,0)*C(0+9,6) = C(6,0)*C(9,6) = 1*84 = 84.
If n=6 then C(6+6,6)*C(6+9,6) = C(12,6)*C(15,6) = 924*5005 = 4624620.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Crossrefs

Cf. A062145.

Programs

Magma

A105942:= func< n | Binomial(n+6,6)*Binomial(n+9,6)/42 >;
[A105942(n): n in [0..40]]; // G. C. Greubel, Mar 11 2025

Mathematica

Table[Binomial[n+6,n]Binomial[n+9,6],{n,0,30}] (* or *) CoefficientList[ Series[-((42 (x+1) (x (2 x+7)+2))/(x-1)^13),{x,0,30}],x] (* Harvey P. Dale, Sep 14 2012 *)

SageMath

def A105942(n): return binomial(n+6,6)*binomial(n+9,6)
print([A105942(n) for n in range(41)]) # G. C. Greubel, Mar 11 2025

Formula

G.f.: 42*(1 + x)*(2 + 7*x + 2*x^2)/(1-x)^13. - Harvey P. Dale, Sep 14 2012

a(0)=84, a(1)=1470, a(2)=12936, a(3)=77616, a(4)=360360, a(5)=1387386, a(6)=4624620, a(7)=13741728, a(8)=37165128, a(9)=92912820, a(10)=217273056, a(11)=479693760, a(12)=1007356896, a(n) = 13*a(n-1) -78*a(n-2) +286*a(n-3) -715*a(n-4) +1287*a(n-5) -1716*a(n-6) +1716*a(n-7) -1287*a(n-8) +715*a(n-9) -286*a(n-10) +78*a(n-11) -13*a(n-12) +a(n-13). - Harvey P. Dale, Sep 14 2012

From Amiram Eldar, Sep 08 2022: (Start)

Sum_{n>=0} 1/a(n) = 10446039/3920 - 270*Pi^2.

Sum_{n>=0} (-1)^n/a(n) = 82911/560 - 15*Pi^2. (End)

Extensions

Corrected and extended by Harvey P. Dale, Sep 14 2012