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A350893
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Number of partitions of n such that (smallest part) = 2*(number of parts).
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5
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0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 10, 10, 12, 13, 15, 16, 19, 20, 23, 25, 28, 30, 34, 36, 40, 43, 47, 50, 56, 59, 65, 70, 77, 82, 91, 97, 107, 115, 126, 135, 149, 159, 174, 187, 204, 218, 238, 254, 276, 295, 320, 341, 370, 394, 426, 455, 491, 523, 565
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OFFSET
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1,18
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} x^(2*k^2)/Product_{j=1..k-1} (1-x^j).
a(n) ~ (1 - alfa) * exp(2*sqrt(n*(2*log(alfa)^2 + polylog(2, 1 - alfa)))) * (2*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(4 - 3*alfa) * n^(3/4)), where alfa = 0.72449195900051561158837228218703656578649448135... is positive real root of the equation alfa^4 + alfa - 1 = 0. - Vaclav Kotesovec, Jan 21 2022
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MATHEMATICA
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nmax = 100; Rest[CoefficientList[1 + Series[Sum[x^(2*j^2)*(1 - x^j)/Product[1 - x^i, {i, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 21 2022 *)
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PROG
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(PARI) my(N=99, x='x+O('x^N)); concat(0, Vec(sum(k=1, sqrtint(N\2), x^(2*k^2)/prod(j=1, k-1, 1-x^j))))
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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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