A053135 Binomial coefficients C(2*n+6,6).

1, 28, 210, 924, 3003, 8008, 18564, 38760, 74613, 134596, 230230, 376740, 593775, 906192, 1344904, 1947792, 2760681, 3838380, 5245786, 7059052, 9366819, 12271512, 15890700, 20358520, 25827165, 32468436, 40475358, 50063860, 61474519, 74974368, 90858768

Offset: 0

Author

Wolfdieter Lang

Keywords

• nonn
• easy

Comments

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

Number of standard tableaux of shape (2n+1,1^6). - Emeric Deutsch, May 30 2004

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..1000
• Milan Janjić, Enumerative Formulas for Some Functions on Finite Sets.
• Milan Janjić, Two Enumerative Functions.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000384, A000579, A002299, A007318, A053128, A053134, A190152.

Programs

Magma

[Binomial(2*n+6, 6): n in [0..30]]; // G. C. Greubel, Sep 03 2018

Maple

seq(binomial(2*n+6,6),n=0..40); # Nathaniel Johnston, May 14 2011

Mathematica

Table[Binomial[2*n+6, 6], {n, 0, 30}] (* G. C. Greubel, Sep 03 2018 *)

PARI

vector(30,n,n--; binomial(2*n+6, 6)) \\ G. C. Greubel, Sep 03 2018

Formula

G.f.: (1 + 21*x + 35*x^2 + 7*x^3)/(1-x)^7.

a(n) = binomial(2*n+6, 6) = A000579(2*n+6).

a(n) = A000384(n+1)*A000384(n+2)*A000384(n+3)/90. - Bruno Berselli, Nov 12 2014

E.g.f.: (90 + 2430*x + 6975*x^2 + 5655*x^3 + 1710*x^4 + 204*x^5 + 8*x^6)* exp(x)/90. - G. C. Greubel, Sep 03 2018

From Amiram Eldar, Oct 21 2022: (Start)

Sum_{n>=0} 1/a(n) = 96*log(2) - 131/2.

Sum_{n>=0} (-1)^n/a(n) = 23/2 - 6*Pi + 12*log(2). (End)