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A265257
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Number of odd singletons in all partitions of n (n>=0).
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3
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0, 1, 0, 2, 2, 5, 5, 11, 13, 23, 28, 45, 57, 86, 108, 156, 199, 276, 350, 475, 601, 798, 1005, 1312, 1646, 2120, 2643, 3365, 4178, 5264, 6500, 8122, 9981, 12375, 15136, 18638, 22697, 27779, 33679, 40993, 49504, 59947, 72109
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OFFSET
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0,4
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LINKS
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FORMULA
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G.f.: g(x) = x(1 - x + x^2)/((1-x^4)*Product_{j>=1}(1-x^j)).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*Pi*sqrt(2*n)).
(End)
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EXAMPLE
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a(6) = 5 because in [1,1,1,3], [1,2,3], [1,5] we have 1+2+2 odd singletons, while the other 8 partitions of 6 have no odd singletons.
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MAPLE
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g := x*(1-x+x^2)/((1-x^4)*mul(1-x^j, j = 1 .. 80)): gser := series(g, x = 0, 55): seq(coeff(gser, x, m), m = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
`if`(i<1, 0, add((p-> `if`(j=1 and i::odd, p+
[0, p[1]], p))(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> b(n$2)[2]:
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MATHEMATICA
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nmax = 50; CoefficientList[Series[x*(1-x+x^2)/(1-x^4) * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 01 2016 *)
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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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