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A339417 Number of compositions (ordered partitions) of n into an odd number of triangular numbers.

%I A339417 #7 Dec 03 2020 18:14:36
%S A339417 0,1,0,2,0,4,1,9,3,19,12,41,33,91,92,203,238,466,602,1080,1493,2536,
%T A339417 3661,6001,8902,14278,21554,34094,52013,81602,125297,195582,301475,
%U A339417 469193,724881,1126161,1742206,2703888,4186276,6493192,10057553,15594636,24161364,37455851
%N A339417 Number of compositions (ordered partitions) of n into an odd number of triangular numbers.
%H A339417 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%H A339417 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F A339417 G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k*(k + 1)/2)) - 1 / Sum_{k>=0} x^(k*(k + 1)/2)).
%F A339417 a(n) = (A023361(n) - A106507(n)) / 2.
%F A339417 a(n) = -Sum_{k=0..n-1} A023361(k) * A106507(n-k).
%e A339417 a(8) = 3 because we have [6, 1, 1], [1, 6, 1] and [1, 1, 6].
%p A339417 b:= proc(n, t) option remember; local r, f, g;
%p A339417       if n=0 then t else r, f, g:=$0..2; while f<=n
%p A339417       do r, f, g:= r+b(n-f, 1-t), f+g, g+1 od; r fi
%p A339417     end:
%p A339417 a:= n-> b(n, 0):
%p A339417 seq(a(n), n=0..50);  # _Alois P. Heinz_, Dec 03 2020
%t A339417 nmax = 43; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(k (k + 1)/2), {k, 1, nmax}]) - 1/Sum[x^(k (k + 1)/2), {k, 0, nmax}]), {x, 0, nmax}], x]
%Y A339417 Cf. A000217, A023361, A106507, A166444, A339374, A339416.
%K A339417 nonn
%O A339417 0,4
%A A339417 _Ilya Gutkovskiy_, Dec 03 2020