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

0,3

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).

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)

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 *)

Prog

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

Crossrefs

Sequence in context: A083575 A302869 A170880 * A046616 A155449 A220154

Adjacent sequences: A239764 A239765 A239766 * A239768 A239769 A239770

Keyword

nonn,easy

Author

Eric Werley, Mar 26 2014

Extensions

More terms from Colin Barker, Mar 26 2014

Status

approved