A053134 Binomial coefficients C(2*n+4,4).

1, 15, 70, 210, 495, 1001, 1820, 3060, 4845, 7315, 10626, 14950, 20475, 27405, 35960, 46376, 58905, 73815, 91390, 111930, 135751, 163185, 194580, 230300, 270725, 316251, 367290, 424270, 487635, 557845, 635376, 720720, 814385, 916895, 1028790, 1150626, 1282975

Offset: 0

Author

Wolfdieter Lang

Keywords

• nonn
• easy

Comments

Even-indexed members of fifth column of Pascal's triangle A007318.

Number of standard tableaux of shape (2n+1,1^4). - Emeric Deutsch, May 30 2004

Number of integer solutions to -n <= x <= y <= z <= w <= n. - Michael Somos, Dec 28 2011

Examples

			1 + 15*x + 70*x^2 + 210*x^3 + 495*x^4 + 1001*x^5 + 1820*x^6 + 3060*x^7 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Milan Janjić, Two Enumerative Functions.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000447, A002299, A053126, A000332.

Programs

Magma

[Binomial(2*n+4,4): n in [0..30]]; // Vincenzo Librandi, Oct 07 2011

Mathematica

Table[Binomial[2*n+4,4], {n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5 ,1}, {1, 15, 70, 210, 495}, 30] (* G. C. Greubel, Sep 03 2018 *)

PARI

for(n=0,30, print1(binomial(2*n+4,4), ", ")) \\ G. C. Greubel, Sep 03 2018

Formula

a(n) = binomial(2*n+4, 4) = A000332(2*n+4).

G.f.: (1 + 10*x + 5*x^2)/(1-x)^5.

a(1 - n) = A053126(n). - Michael Somos, Dec 28 2011

E.g.f.: (6 + 84*x + 123*x^2 + 44*x^3 + 4*x^4)*exp(x)/6. - G. C. Greubel, Sep 03 2018

a(n) = (1/6)*(n + 1)*(n + 2)*(2*n + 1)*(2*n + 3). - Gerry Martens, Oct 13 2022

From Amiram Eldar, Oct 21 2022: (Start)

Sum_{n>=0} 1/a(n) = 16*log(2) - 10.

Sum_{n>=0} (-1)^n/a(n) = 10 - 2*Pi - 4*log(2). (End)