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

%I A126683 #15 May 25 2018 03:07:37
%S A126683 1,1,1,1,2,4,8,16,33,68,144,312,686,1523,3405,7652,17284,39246,89552,
%T A126683 205253,472297,1090544,2525904,5867037,13663248,31896309,74628130,
%U A126683 174972341,411032475,967307190,2280248312,5383723722,12729879673,30141755384,71462883813
%N A126683 Number of partitions of the n-th triangular number n(n+1)/2 into distinct odd parts.
%C A126683 Also the number of self-conjugate partitions of the n-th triangular number.
%H A126683 Alois P. Heinz, <a href="/A126683/b126683.txt">Table of n, a(n) for n = 0..200</a>
%e A126683 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.
%p A126683 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
%p A126683 # second Maple program:
%p A126683 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i^2<n, 0,
%p A126683       b(n, i-1)+`if`(2*i-1>n, 0, b(n-2*i+1, i-1))))
%p A126683     end:
%p A126683 a:= n-> b(n*(n+1)/2, ceil(n*(n+1)/4)*2-1):
%p A126683 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jan 31 2018
%t A126683 a[n_] := SeriesCoefficient[QPochhammer[-x, x^2], {x, 0, n*(n+1)/2}];
%t A126683 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, May 25 2018 *)
%Y A126683 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.
%Y A126683 Cf. A000217.
%K A126683 nonn
%O A126683 0,5
%A A126683 _Moshe Shmuel Newman_, Feb 15 2007
%E A126683 More terms from _Emeric Deutsch_, Feb 27 2007
%E A126683 a(0)=1 prepended by _Alois P. Heinz_, Jan 31 2018