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A053261
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Coefficients of the '5th-order' mock theta function psi_1(q).
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17
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1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 10, 10, 10, 11, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 19, 20, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 34, 35, 37, 39, 40, 41, 44, 45, 47, 50, 51, 53, 56, 58, 60, 63, 65
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OFFSET
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0,7
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COMMENTS
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Number of partitions of n such that each part occurs at most twice and if k occurs as a part then all smaller positive integers occur.
Strictly unimodal compositions with rising range 1, 2, 3, ..., m where m is the largest part and distinct parts in the falling range (this follows trivially from the comment above). [Joerg Arndt, Mar 26 2014]
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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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22.
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LINKS
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FORMULA
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G.f.: psi_1(q) = Sum_{n>=0} q^(n*(n+1)/2) * Product_{k=1..n} (1 + q^k).
a(n) ~ sqrt(phi) * exp(Pi*sqrt(n/15)) / (2*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i>n, 0, add(b(n-i*j, i+1), j=1..min(2, n/i))))
end:
a:= n-> b(n, 1):
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MATHEMATICA
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Series[Sum[q^(n(n+1)/2) Product[1+q^k, {k, 1, n}], {n, 0, 13}], {q, 0, 100}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, Sum[b[n - i*j, i + 1], {j, 1, Min[2, n/i]}]]];
a[n_] := b[n, 1];
nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2) * Product[1+x^j, {j, 1, k}], {k, 0, Floor[Sqrt[2*nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)
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PROG
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(PARI) N = 66; x = 'x + O('x^N); gf = sum(n=0, N, x^(n*(n+1)/2) * prod(k=1, n, 1+x^k) ); v = Vec(gf) /* Joerg Arndt, Apr 21 2013 */
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CROSSREFS
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Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053262, 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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