A107417 a(n) = binomial(n+2,2)*binomial(n+5,5).

1, 18, 126, 560, 1890, 5292, 12936, 28512, 57915, 110110, 198198, 340704, 563108, 899640, 1395360, 2108544, 3113397, 4503114, 6393310, 8925840, 12273030, 16642340, 22281480, 29484000, 38595375, 50019606, 64226358, 81758656, 103241160, 129389040, 161017472, 199051776

Offset: 0

Author

Zerinvary Lajos, May 26 2005

Keywords

• easy
• nonn

Examples

			If n=0 then C(0+2,2)*C(0+5,5) = C(2,2)*C(5,5) = 1*1 = 1.
If n=3 then C(3+2,2)*C(3+5,5) = C(5,2)*C(8,5) = 10*56 = 560.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Crossrefs

Cf. A062145.

Programs

Magma

A107417:= func< n | Binomial(n+2,n)*Binomial(n+5,n) >;
[A107417(n): n in [0..40]]; // G. C. Greubel, Mar 10 2025

Mathematica

Table[Binomial[n+2,2]Binomial[n+5,5],{n,0,40}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{1,18,126,560,1890,5292,12936,28512},40] (* Harvey P. Dale, Feb 18 2012 *)

PARI

for(n=0,40,print1(binomial(n+2,2)*binomial(n+5,5),","))

SageMath

def A107417(n): return binomial(n+2,n)*binomial(n+5,n)
print([A107417(n) for n in range(41)]) # G. C. Greubel, Mar 10 2025

Formula

From Harvey P. Dale, Feb 18 2012: (Start)

a(0)=1, a(1)=18, a(2)=126, a(3)=560, a(4)=1890, a(5)=5292, a(6)=12936, a(7)=28512, a(n)=8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)- 28*a(n-6)+ 8*a(n-7)-a(n-8).

G.f.: (1 + 10*x + 10*x^2)/(1-x)^8. (End)

From Amiram Eldar, Sep 08 2022: (Start)

Sum_{n>=0} 1/a(n) = 25*Pi^2/3 - 5845/72.

Sum_{n>=0} (-1)^n/a(n) = 205/8 - 5*Pi^2/2. (End)

E.g.f.: (1/240)*(240 + 4080*x + 10920*x^2 + 9400*x^3 + 3350*x^4 + 542*x^5 + 39*x^6 + x^7)*exp(x). - G. C. Greubel, Mar 10 2025

Extensions

More terms from Rick L. Shepherd, May 27 2005