A126683 Number of partitions of the n-th triangular number n(n+1)/2 into distinct odd parts.

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

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^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):
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: A182442 A098588 A367715 * 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