A239767 - OEIS

%I #26 Feb 26 2022 12:30:57

%S 0,1,6,11,22,31,48,61,84,101,130,151,186,211,252,281,328,361,414,451,

%T 510,551,616,661,732,781,858,911,994,1051,1140,1201,1296,1361,1462,

%U 1531,1638,1711,1824,1901,2020,2101,2226,2311,2442,2531,2668,2761,2904,3001

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

%C 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]

%C 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.

%H Vincenzo Librandi, <a href="/A239767/b239767.txt">Table of n, a(n) for n = 0..1000</a>

%H George E. Andrews, <a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC348728/">The hard-hexagon model and Rogers-Ramanujan type identities</a>, Proc. Nat. Acad. Sci. U.S.A., 78(1981), 5290-5292.

%H L. J. Slater, <a href="http://plms.oxfordjournals.org/content/s2-54/1/147.extract">Further Identities of the Rogers-Ramanujan Type</a>, Proc. London Math. Soc., 54(1952), 147-167.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

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

%F From _Colin Barker_, Mar 26 2014: (Start)

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

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

%p 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

%t Table[(10 n^2 + 2 n (1 + (-1)^n) - (1 - (-1)^n))/8, {n, 0, 100}] (* _Wesley Ivan Hurt_, Mar 27 2014 *)

%t 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 *)

%o (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

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

%K nonn,easy

%O 0,3

%A _Eric Werley_, Mar 26 2014

%E More terms from _Colin Barker_, Mar 26 2014