%I #29 Sep 20 2022 02:38:13
%S 1,42,273,969,2530,5481,10472,18278,29799,46060,68211,97527,135408,
%T 183379,243090,316316,404957,511038,636709,784245,956046,1154637,
%U 1382668,1642914,1938275,2271776,2646567,3065923,3533244,4052055,4626006,5258872,5954553,6717074
%N a(n) = binomial(5*(n+1),4)/5, with n >= 0.
%C Used as one of the 5-section parts of A234042.
%C 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
%H Vincenzo Librandi, <a href="/A234043/b234043.txt">Table of n, a(n) for n = 0..1000</a>
%H Alison Schuetz and Gwyneth Whieldon, <a href="http://arxiv.org/abs/1401.7194">Polygonal Dissections and Reversions of Series</a>, arXiv:1401.7194 [math.CO], 2014.
%F G.f: (1 + 37*x + 73*x^2 + 14*x^3)/(1-x)^5.
%F a(n) = A234042(5*n+1) for n >= 0.
%F a(n) = (n+1)*(5*n+2)*(5*n+3)*(5*n+4)/24.
%F From _Amiram Eldar_, Sep 20 2022: (Start)
%F 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).
%F Sum_{n>=0} (-1)^n/a(n) = 4*sqrt(5)*log(phi) + 2*sqrt(26-38/sqrt(5))*Pi - 32*log(2). (End)
%t CoefficientList[Series[(1 + 37 x + 73 x^2 + 14 x^3)/(1 - x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 26 2014 *)
%o (Magma) [Binomial(5*(n+1),4)/5: n in [0..40]]; // _Vincenzo Librandi_, Feb 26 2014
%Y Cf. A100157, A000326, A001622, A151989, A234042.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Feb 24 2014