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

Author

Moshe Shmuel Newman, Feb 15 2007

Keywords

• nonn

Comments

Also the number of self-conjugate partitions of the n-th triangular number.

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

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.

Programs

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 *)

Extensions

More terms from Emeric Deutsch, Feb 27 2007

a(0)=1 prepended by Alois P. Heinz, Jan 31 2018