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A264591
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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[4](q).
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9
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1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 20, 22, 26, 28, 33, 36, 42, 46, 53, 58, 67, 73, 83, 91, 104, 113, 128, 140, 158, 173, 194, 212, 238, 260, 290, 317, 353, 385, 428, 467, 517, 564
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OFFSET
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0,11
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COMMENTS
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It is conjectured that G[i](q) = 1 + O(q^i) for all i.
For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[4](x). - N. J. A. Sloane, Nov 22 2015
The second g.f. given below leads to a combinatorial partition interpretation from (2 + 4 + ... + 2*m) + 2*m = m*(m+3). Take for the sum term m the special M=m+1 part partition [2m,2m,2*(m-1),...,4,2] together with arbitrary partitions of N with part number m' <= M-1 = m added to the first m' parts.
Summing over m>=1 leads to partitions of n = m*(m+3) + N which have no part 1, only one part 2 except for n=4 and for number of parts M >= 3 the difference of parts except of the first two parts has to be at least 2. See the examples below.
A simpler interpretation uses m*(m+3) = 4 + 6 + ... + 2*(m+1), leading to a(n) as the number of partitions of n with parts >= 4 and parts differing by at least 2.
This is in the spirit of MacMahon's and Schur's interpretation of the sum version of the Rogers-Ramanujan identities. See the Hardy and Hardy-Wright references under A003114. (End)
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LINKS
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FORMULA
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G.f.: G[4](q) = Sum_{n >= 0} (-1)^n*(1 - q^(n+1))*(1 - q^(n+2))*(1 - q^(2*n+3)) * q^((5*n+11)*n/2)/Product_{j >= 1} (1 - q^j)), from the Andrews-Baxter (AB) reference, eq. (3.7).
G.f.: Sum_{m >= 0} q^(m*(m+3)) / Product_{j=1..m} (1-q^j) from (AB) eq. 51.
This can also be derived from the Hardy (H) or Hardy-Wright reference (see A006141): Put G_4(a,q):= (H_1(a,q) - H_1(a*q,q)) / (a*q) with H_1(a,x) from (H) p. 95, first eq. Then G[4](q) = G_4(q,q). (End)
a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * sqrt(5) * phi^(5/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Dec 24 2016
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EXAMPLE
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a(0) = 1 from the n=0 sum term (undefined product put to 1),
a(n) = 1 for n=4..9 from the partition [n-2,2],
a(10) = 2 from [8,2] and [4,4,2],
a(11) = 2 from [9,2] and [5,4,2],
a(12) = 3 from [10,2], [6,4,2], [5,5,2],
a(18) = 7 from [16,2], all 1+4=5 partitions of 18-10 = 8 with part number <= 2 added to the first two part of [4,4,2] and the new four part partition [6,6,4,2].
The maximal number of parts needed for n is floor((-1+sqrt(9+4*n))/2) = A259361(n+2).
A simpler interpretation:
a(18) = 7 from the partitions of 18 with parts >=4 and parts differing by at least 2: [18], [14,4], [13,5], [12,6], [11,7], [10,8], [8,6,4].
The maximal number of parts needed for n is floor((-3+sqrt(9+4*n))/2).
(End)
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MATHEMATICA
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nmax = 100; CoefficientList[Series[Sum[x^(k*(k+3))/Product[1-x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 24 2016 *)
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CROSSREFS
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For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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