0,2

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.

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.

Colin Barker, Table of n, a(n) for n = 0..1000

Bryan Ek, q-Binomials and related symmetric unimodal polynomials, arXiv:1711.11252 [math.CO], 2017-2018.

Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

From Colin Barker, Apr 11 2018: (Start)

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

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. (End)

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.

A302612[n_] := (n+1)*((n-1)*n*((n-3)*n + 8)/12 + 1); Array[A302612, 50, 0] (* or *)

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 2, 6, 20, 65, 186}, 50] (* Paolo Xausa, Feb 26 2026 *)

(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

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

Sequence in context: A220874 A273902 A181301 * A005726 A148473 A000718

Adjacent sequences: A302609 A302610 A302611 * A302613 A302614 A302615

nonn,easy

Bryan T. Ek, Apr 10 2018

More terms from Colin Barker, Apr 11 2018

approved