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A276434
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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, 0, 1, 1, 3, 3, 5, 6, 10, 12, 19, 23, 34, 41, 58, 72, 98, 121, 162, 200, 262, 323, 415, 511, 650, 796, 1000, 1222, 1522, 1851, 2287, 2771, 3399, 4103, 5000, 6015, 7289, 8735, 10530, 12579, 15094, 17968, 21468, 25477, 30319, 35873, 42531, 50177, 59291
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OFFSET
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0,7
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000
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FORMULA
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a(n) = Sum(k*A276433(n,k), k>=0).
G.f.: g(x) = Sum_(i>=1) (x^(i(i+1))(1-x^i))/Product_(i>=1) (1-x^i).
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EXAMPLE
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a(6) = 3 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), 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]:
seq(a(n), n=0..60); # Alois P. Heinz, Sep 30 2016
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MATHEMATICA
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max = 60; s = Sum[x^(i*(i+1))*(1-x^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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Cf. A276427, A276428, A276429, A276433, A277099, A277100, A277101, A277102.
Sequence in context: A059892 A241090 A091607 * A183561 A300183 A222704
Adjacent sequences: A276431 A276432 A276433 * A276435 A276436 A276437
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch, Sep 30 2016
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STATUS
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approved
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