A234043 a(n) = binomial(5*(n+1),4)/5, with n >= 0.

1, 42, 273, 969, 2530, 5481, 10472, 18278, 29799, 46060, 68211, 97527, 135408, 183379, 243090, 316316, 404957, 511038, 636709, 784245, 956046, 1154637, 1382668, 1642914, 1938275, 2271776, 2646567, 3065923, 3533244, 4052055, 4626006, 5258872, 5954553, 6717074

Offset: 0

Author

Wolfdieter Lang, Feb 24 2014

Keywords

• nonn
• easy

Comments

Used as one of the 5-section parts of A234042.

The Fuss-Catalan numbers are Cat(d,k) = (1/(k*(d-1)+1))*binomial(k*d,k) and enumerate the (d+1)-gon partitions of a (k*(d-1)+2)-gon (cf. Whieldon and Schuetz link). a(n) = Cat(n,5) (Offset=1), so enumerates the (n+1)-gon partitions of a (5*(n-1)+2)-gon. Analogous series are A000326 (k=3) and A100157 (k=4). - Tom Copeland, Oct 05 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.

Crossrefs

Cf. A100157, A000326, A001622, A151989, A234042.

Programs

Magma

[Binomial(5*(n+1),4)/5: n in [0..40]]; // Vincenzo Librandi, Feb 26 2014

Mathematica

CoefficientList[Series[(1 + 37 x + 73 x^2 + 14 x^3)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)

Formula

G.f: (1 + 37*x + 73*x^2 + 14*x^3)/(1-x)^5.

a(n) = A234042(5*n+1) for n >= 0.

a(n) = (n+1)*(5*n+2)*(5*n+3)*(5*n+4)/24.

From Amiram Eldar, Sep 20 2022: (Start)

Sum_{n>=0} 1/a(n) = 10*sqrt(5)*log(phi) + 5*log(5) - 2*sqrt(25-38/sqrt(5))*Pi, where phi is the golden ratio (A001622).

Sum_{n>=0} (-1)^n/a(n) = 4*sqrt(5)*log(phi) + 2*sqrt(26-38/sqrt(5))*Pi - 32*log(2). (End)