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A277101
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Sum over all partitions of n of the number of distinct parts i of multiplicity i - 1.
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8
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0, 0, 1, 1, 1, 2, 4, 5, 8, 10, 15, 20, 29, 37, 52, 67, 89, 115, 152, 192, 251, 316, 405, 508, 644, 799, 1006, 1243, 1546, 1901, 2351, 2871, 3527, 4289, 5232, 6336, 7688, 9264, 11189, 13430, 16137, 19299, 23097, 27514, 32799, 38944, 46246, 54738, 64782, 76430, 90171
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OFFSET
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0,6
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LINKS
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FORMULA
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G.f.: g(x) = Sum_(i>=1)(x^(i(i+1))(1-x^(i+1)))/Product_(i>=1)(1-x^i).
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EXAMPLE
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a(6) = 4 because in the partitions [1,1,1,1,1,1], [1,1,1,1,2'], [1,1,2,2], [2,2,2], [1,1,1,3], [1,2',3], [3',3], [1,1,4], [2',4], [1,5], [6] of 6 only the marked parts satisfy the requirement.
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MAPLE
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g := (sum(x^(i*(i+1))*(1-x^(i+1)), i = 1 .. 200))/(product(1-x^i, i = 1 .. 200)): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
`if`(i<1, 0, add((p-> p+`if`(i-1<>j, 0,
[0, p[1]]))(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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max = 60; s = Sum[x^(i*(i+1))*(1-x^(1+i)), {i, 1, max}]/QPochhammer[x] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 08 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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