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 A126683 Number of partitions of the n-th triangular number n(n+1)/2 into distinct odd parts. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Also the number of self-conjugate partitions of the n-th triangular number. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..200 EXAMPLE 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. MAPLE 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^2n, 0, b(n-2*i+1, i-1))))     end: a:= n-> b(n*(n+1)/2, ceil(n*(n+1)/4)*2-1): seq(a(n), n=0..40);  # Alois P. Heinz, Jan 31 2018 MATHEMATICA a[n_] := SeriesCoefficient[QPochhammer[-x, x^2], {x, 0, n*(n+1)/2}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 25 2018 *) CROSSREFS 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. Cf. A000217. Sequence in context: A324406 A182442 A098588 * A005821 A177794 A004149 Adjacent sequences:  A126680 A126681 A126682 * A126684 A126685 A126686 KEYWORD nonn AUTHOR Moshe Shmuel Newman, Feb 15 2007 EXTENSIONS More terms from Emeric Deutsch, Feb 27 2007 a(0)=1 prepended by Alois P. Heinz, Jan 31 2018 STATUS approved

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Last modified May 12 05:25 EDT 2021. Contains 343811 sequences. (Running on oeis4.)