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A126683
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Number of partitions of the n-th triangular number n(n+1)/2 into distinct odd parts.
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4
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1, 1, 1, 1, 2, 4, 8, 16, 33, 68, 144, 312, 686, 1523, 3405, 7652, 17284, 39246, 89552, 205253, 472297, 1090544, 2525904, 5867037, 13663248, 31896309, 74628130, 174972341, 411032475, 967307190, 2280248312, 5383723722, 12729879673, 30141755384, 71462883813
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OFFSET
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0,5
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COMMENTS
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Also the number of self-conjugate partitions of the n-th triangular number.
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LINKS
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EXAMPLE
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The 5th triangular number is 15. Writing this as a sum of distinct odd numbers: 15 = 11 + 3 + 1 = 9 + 5 + 1 = 7 + 5 + 3 are all the possibilities. So a(5) = 4.
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MAPLE
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g:= mul(1+x^(2*j+1), j=0..900): seq(coeff(g, x, n*(n+1)/2), n=0..40); # Emeric Deutsch, Feb 27 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i^2<n, 0,
b(n, i-1)+`if`(2*i-1>n, 0, b(n-2*i+1, i-1))))
end:
a:= n-> b(n*(n+1)/2, ceil(n*(n+1)/4)*2-1):
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MATHEMATICA
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a[n_] := SeriesCoefficient[QPochhammer[-x, x^2], {x, 0, n*(n+1)/2}];
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CROSSREFS
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Sequences A066655 and A104383 do the same thing for triangular numbers, with partitions or distinct partitions. Sequences A072213 and A072243 are analogs for squares rather than triangular numbers.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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