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A143184
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Coefficients of a Ramanujan q-series.
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7
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1, 1, 2, 4, 6, 10, 15, 23, 33, 49, 69, 98, 136, 188, 256, 348, 466, 622, 824, 1084, 1418, 1846, 2389, 3077, 3947, 5038, 6407, 8115, 10241, 12876, 16141, 20160, 25110, 31179, 38609, 47674, 58724, 72141, 88421, 108114, 131902, 160565, 195061, 236468
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OFFSET
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0,3
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COMMENTS
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Also equal to the number of overpartitions of n with no non-overlined parts larger than the number of overlined parts. For example, the overpartitions counted by a(4) = 6 are: [4'], [3',1], [3',1'], [2',1,1], [2',1',1], [1',1,1,1]. - Jeremy Lovejoy, Aug 23 2021
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REFERENCES
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S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 10
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LINKS
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FORMULA
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G.f.: Sum_{k>=0} x^((k^2+k)/2) / ((1 - x) * (1 - x^2) ... (1 - x^k))^2.
a(n) ~ exp(2*Pi*sqrt(n/5)) / (2^(3/2) * 5^(3/4) * n). - Vaclav Kotesovec, Nov 20 2020
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EXAMPLE
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G.f. = 1 + q + 2*q^2 + 4*q^3 + 6*q^4 + 10*q^5 + 15*q^6 + 23*q^7 + 33*q^8 + ...
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MAPLE
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b:= proc(n, i) option remember;
`if`(i>n, 0, `if`(irem(n, i, 'r')=0, r, 0)+
add(j*b(n-i*j, i+1), j=1..n/i))
end:
a:= n-> `if`(n=0, 1, b(n, 1)):
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MATHEMATICA
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m = 50;
Sum[x^(k(k+1)/2)/Product[1-x^j, {j, 1, k}]^2, {k, 0, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 20 2020 *)
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PROG
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(PARI) {a(n)= local(t); if( n<0, 0, t = 1 +x*O(x^n); polcoeff( sum(k=1, (sqrtint(8*n+1)-1)\2, t = t* x^k/ (1-x^k)^2 +x*O(x^n), 1), n))}
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CROSSREFS
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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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