%I #22 Dec 24 2016 10:29:48

%S 1,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,6,7,8,9,11,12,14,16,18,20,23,

%T 25,29,32,36,40,46,50,57,63,71,78,88,96,108,119,132,145,162,177,197,

%U 216,239,262,290,317,350,383,421,460,507,552,606,661,724,789,864,939,1027,1117

%N Let G[1](q) denote the g.f. for A003114 and G[2](q) the g.f. for A003106 (the two Rogers-Ramanujan identities). For i>=3, let G[i](q) = (G[i-1](q)-G[i-2](q))/q^(i-2). Sequence gives coefficients of G[5](q).

%C It is conjectured that G[i](q) = 1 + O(q^i) for all i.

%C From _Wolfdieter Lang_, Nov 03 2016: (Start)

%C The generalized Rogers-Ramanujan [R-R] series G[i](q) of Andrews and Baxter [A-B] have a standard combinatorial interpretation of the Schur and MacMahon type (see Hardy [H] and Hardy-Wright [H-W] for the original [R-R] case) inferred from the formula G[i](q) = Sum_{m >=0} q^(m*(m+i-1)}/Product_{j =1..m} (1 - q^j) ([A-B], eq. (5.1)). Define GI_k(q) = G[2*k+1](q) and GII_k(q) = G[2*k](q), for k = 0, 1,..., and prove the two formulas I(m,k): m*(m+2*k) = Sum_{j = 1..2*m-1} (2*k + j), and II(m,k): m*(m+2*k+1) = Sum_{j = 1..m} (2*(k + j)) for fixed positive m by induction over k = 0, 1, ... . For GI_k(q) define the special m-part partition SPI(m,k) = [2*k+2*m-1,2*k+2*m-3,...,2*k+1] of m*(m+2*k), and for GII_k(q) the special m-part partition SPII(m,k) [2*(k+1),2*(k+2),...,2*(k+1))] of m*(m+2*k+1).

%C Then GI_k(q) = 1 + Sum_{n >=1} aI(k,n)*q^n with aI(k,n) the number of partitions of n without parts 1, 2, ..., 2*k, and the parts differ by at least 2. GII_k(q) = 1 + Sum_{n >=1} aII(k,n)*q^n with aII(k,n) the number of partitions of n without parts 1, 2, ..., 2*k+1, and the parts differ by at least 2. The proof can be directly adapted from the one given in [H] or [H-W] for k=1.

%C For the partitions of n generated by GI_k(q) one needs the maximal part number MmaxI(k,n) = floor(-k + sqrt(k^2 + n)). For the GII_k(q) case MmaxII(k,n) = floor(-(2*k+1) + sqrt((2*k+1)^2 + 4*n)).

%C The present sequence is aI(2,n), in [A-B] notation generated by G[5](q), giving the number of partitions of n without parts 1, 2, 3 and 4, and the parts differ by at least 2.

%C (End)

%D G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 91-92.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, pp. 290-291.

%H Vaclav Kotesovec, <a href="/A264592/b264592.txt">Table of n, a(n) for n = 0..1000</a>

%H George E. Andrews; R. J. Baxter, <a href="http://www.computing-wisdom.com/jstor/rogers-ramanujan.pdf">A motivated proof of the Rogers-Ramanujan identities</a>, Amer. Math. Monthly 96 (1989), no. 5, 401-409.

%H Shashank Kanade, <a href="http://www.math.rutgers.edu/~skanade/SK-Defense-Handout.pdf">Some results on the representation theory of vertex operator algebras and integer partition identities</a>, PhD Handout, Math. Dept., Rutgers University, April 2015.

%H Shashank Kanade, <a href="http://dx.doi.org/doi:10.7282/T3TX3H7B">Some results on the representation theory of vertex operator algebras and integer partition identities</a>, PhD Dissertation, Math. Dept., Rutgers University, April 2015.

%F From _Wolfdieter Lang_, Nov 03 2016: (Start)

%F G.f.: G[5](q) = GI_2(q) = Sum_{m >=0} q^(m*(m+4)) / Product_{j =1..m} (1 - q^j).

%F See [A-B], eq. (5.1) for i=5.

%F a(0) = 1 and a(n) gives the number of partitions of n without part 1 and 2, the parts differing by at least 2.

%F G.f.: Sum_{m=0} ((-1)^m*(1 - q^(m+1))*(1 - q^(m+2))*(1 - q^(m+3))*(1 - q^(2*(m+2))) * q^(5*(n+3)*n/2)) / Product_{j>=1} (1 - q^j). See [A-B], eq. (3.8) for i=5. (End)

%F a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * sqrt(5) * phi^(7/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Dec 24 2016

%e From _Wolfdieter Lang_, Nov 03 2016: (Start)

%e a(5) = 1 because the only partition of n = 5 without parts 1, 2, 3 and 4, and parts differing by at least 2 is [5].

%e a(12) = 2 from the two partitions [12] and [7,5] of n = 12.

%e a(18) = 5 from the five partitions [18], [13,5], [12,6], [11,7], [10,8] of n = 18.

%e (End)

%t nmax = 100; CoefficientList[Series[Sum[x^(k*(k+4))/Product[1-x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 24 2016 *)

%Y For the generalized Rogers-Ramanujan series G[0], G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003113, A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595.

%K nonn

%O 0,13

%A _N. J. A. Sloane_, Nov 19 2015