%I #25 Jul 12 2018 00:47:46

%S 1,2,6,20,65,186,462,1016,2025,3730,6446,10572,16601,25130,36870,

%T 52656,73457,100386,134710,177860,231441,297242,377246,473640,588825,

%U 725426,886302,1074556,1293545,1546890,1838486,2172512,2553441,2986050,3475430,4026996

%N a(n) = (n+1)*(n^4-4*n^3+11*n^2-8*n+12)/12.

%C The limit as q->1^- of the unimodal polynomial [q^(n*k+n+4)-q^(n*k+n+3)+q^(n*k+n+1)-q^(n*k+4)-q^((n-1)*k+n+3)+q^((n-1)*k+3)+q^(k+n+1)-q^(k+1)-q^n+q^3-q+1]/[(1-q)^2(1-q^2)(1-q^n)] after making the simplification k=n. This unimodal polynomial is from O'Hara's proof of unimodality of q-binomials after making the restriction to partitions of size <=2. See G_2(n,k) from arXiv:1711.11252.

%C As the size restriction s increases, G_s->G_infinity=G: the q-binomials. Then substituting k=n and q=1 yields the central binomial coefficients: A000984.

%H Colin Barker, <a href="/A302612/b302612.txt">Table of n, a(n) for n = 0..1000</a>

%H Bryan Ek, <a href="https://arxiv.org/abs/1711.11252">q-Binomials and related symmetric unimodal polynomials</a>, arXiv:1711.11252 [math.CO], 2017-2018.

%H Bryan Ek, <a href="https://arxiv.org/abs/1804.05933">Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics</a>, arXiv:1804.05933 [math.CO], 2018.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F From _Colin Barker_, Apr 11 2018: (Start)

%F G.f.: (1 - 4*x + 9*x^2 - 6*x^3 + 10*x^4) / (1 - x)^6.

%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5.

%F (End)

%e For n=4, G_2(4,4)=q^16+q^15+2*q^14+3*q^13+5*q^12+5*q^11+6*q^10+6*q^9+7*q^8+6*q^7+6*q^6+5*q^5+5*q^4+3*q^3+2*q^2+q+1 (using the formula in the comments). Then substituting q=1 yields 65.

%o (PARI) Vec((1 - 4*x + 9*x^2 - 6*x^3 + 10*x^4) / (1 - x)^6 + O(x^40)) \\ _Colin Barker_, Apr 11 2018

%Y Cf. A000984, A002522, A302644, A302645, A302646.

%K nonn,easy

%O 0,2

%A _Bryan T. Ek_, Apr 10 2018

%E More terms from _Colin Barker_, Apr 11 2018