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A053262
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Coefficients of the 5th-order mock theta function chi_0(q).
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15
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1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 3, 6, 5, 7, 7, 9, 7, 12, 11, 13, 13, 17, 15, 21, 20, 24, 24, 29, 28, 36, 35, 40, 42, 50, 48, 58, 58, 67, 70, 80, 79, 93, 95, 106, 111, 125, 127, 145, 149, 166, 172, 191, 196, 222, 229, 250, 262, 289, 298, 330, 343, 373, 391, 427, 442, 486
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OFFSET
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0,4
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COMMENTS
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The rank of a partition is its largest part minus the number of parts.
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REFERENCES
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Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 23, 25.
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LINKS
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FORMULA
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G.f.: chi_0(q) = Sum_{n>=0} q^n/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))).
G.f.: chi_0(q) = 1 + Sum_{n>=0} q^(2n+1)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).
a(n) is the number of partitions of 5n with rank == 1 (mod 5) minus number with rank == 0 (mod 5).
a(n) is the number of partitions of n with unique smallest part and all other parts <= twice the smallest part.
a(n) is the number of partitions where the largest part is odd and all other parts are greater than half of the largest part. - N. Sato, Jan 21 2010
a(n) ~ exp(Pi*sqrt(2*n/15)) / sqrt((5 + sqrt(5))*n). - Vaclav Kotesovec, Jun 12 2019
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MATHEMATICA
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1+Series[Sum[q^(2n+1)/Product[1-q^k, {k, n+1, 2n+1}], {n, 0, 49}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[1 + Sum[x^(2*k+1)/Product[1-x^j, {j, k+1, 2*k+1}], {k, 0, Floor[nmax/2]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)
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CROSSREFS
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Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053263, A053264, A053265, A053266, A053267.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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