A105947 a(n) = C(n+4,4) * C(n+6,6).

1, 35, 420, 2940, 14700, 58212, 194040, 566280, 1486485, 3578575, 8016008, 16893240, 33786480, 64574160, 118605600, 210327264, 361499985, 604167795, 984569740, 1568220500, 2446423980, 3744526500, 5632263000, 8336601000, 12157543125, 17487410031, 24834191760

Offset: 0

Author

Zerinvary Lajos, Apr 27 2005

Keywords

• easy
• nonn

Examples

			If n=0 then C(0+6,0)*C(0+4,4) = C(6,0)*C(4,4) = 1*1 = 1.
If n=10 then C(10+6,10)*C(10+4,4) = C(16,10)*C(14,4) = 8008*1001 = 8016008.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Crossrefs

Cf. A062196.

Programs

Magma

A105947:= func< n | Binomial(n+4,4)*Binomial(n+6,6) >;
[A105947(n): n in [0..40]]; // G. C. Greubel, Feb 22 2025

Mathematica

Table[Binomial[n+6,n]Binomial[n+4,4],{n,0,30}] (* Harvey P. Dale, May 21 2014 *)

SageMath

def A105947(n): return binomial(n+4,4)*binomial(n+6,6)
print([A105947(n) for n in range(41)]) # G. C. Greubel, Feb 22 2025

Formula

G.f.: (1 + 24*x + 90*x^2 + 80*x^3 + 15*x^4)/(1-x)^11. - Colin Barker, Jan 28 2013

From Wesley Ivan Hurt, Jan 27 2022: (Start)

a(n) = (17280 + 78336*n + 152376*n^2 + 167780*n^3 + 116150*n^4 + 52983*n^5 +

16173*n^6 + 3270*n^7 + 420*n^8 + 31*n^9 + n^10)/17280.

a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11). (End)

From Amiram Eldar, Sep 08 2022: (Start)

Sum_{n>=0} 1/a(n) = 224*Pi^2 - 55244/25.

Sum_{n>=0} (-1)^n/a(n) = 12*Pi^2 + 512*log(2)/5 - 4711/25. (End)

Extensions

Terms from a(8) onwards corrected by Colin Barker, Jan 28 2013

Second example corrected by Colin Barker, Jan 28 2013