A239767 Degrees of polynomial on the fermionic side of the finite generalization of identity 46 from Slater's List.

0, 1, 6, 11, 22, 31, 48, 61, 84, 101, 130, 151, 186, 211, 252, 281, 328, 361, 414, 451, 510, 551, 616, 661, 732, 781, 858, 911, 994, 1051, 1140, 1201, 1296, 1361, 1462, 1531, 1638, 1711, 1824, 1901, 2020, 2101, 2226, 2311, 2442, 2531, 2668, 2761, 2904, 3001

Offset: 0

Author

Eric Werley, Mar 26 2014

Keywords

• nonn
• easy

Comments

A "Rogers-Ramanujan-Slater" type identity is an identity containing a variable q which equates an infinite product with an infinite series. A finite generalization of such an identity consists of two sequences of polynomials, such that corresponding terms in each sequence are equal and one sequence tends to the infinite sum and the other sequence tends to the infinite product. [From AMS Abstracts 2008 Eric Werley by Michael Somos, Mar 27 2014]

In statistical mechanics, the fermionic side of a Rogers-Ramanujan type identity is the infinite series side of the identity and the bosonic side is the infinite product side of the identity.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• George E. Andrews, The hard-hexagon model and Rogers-Ramanujan type identities, Proc. Nat. Acad. Sci. U.S.A., 78(1981), 5290-5292.
• L. J. Slater, Further Identities of the Rogers-Ramanujan Type, Proc. London Math. Soc., 54(1952), 147-167.
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Programs

Magma

[(1/8)*(10*n^2+2*(1+(-1)^n)*n-(1-(-1)^n)): n in [0..50]]; // Vincenzo Librandi, Mar 29 2014

Maple

A239767:=n->(10*n^2 + 2*n*(1+(-1)^n) - (1-(-1)^n))/8; seq(A239767(n), n=0..100); # Wesley Ivan Hurt, Mar 27 2014

Mathematica

Table[(10 n^2 + 2 n (1 + (-1)^n) - (1 - (-1)^n))/8, {n, 0, 100}] (* Wesley Ivan Hurt, Mar 27 2014 *)
CoefficientList[Series[- x (x^3 + 3 x^2 + 5 x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 29 2014 *)

PARI

concat(0, Vec(-x*(x^3+3*x^2+5*x+1)/((x-1)^3*(x+1)^2) + O(x^100))) \\ Colin Barker, Mar 26 2014

Formula

a(n) = (1/8)*(10*n^2 + 2*(1+(-1)^n)*n - (1-(-1)^n)).

From Colin Barker, Mar 26 2014: (Start)

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

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

Extensions

More terms from Colin Barker, Mar 26 2014