cp's OEIS Frontend

A339416 Number of compositions (ordered partitions) of n into an even number of triangular numbers.

%I A339416 #8 Dec 03 2020 18:15:34
%S A339416 1,0,1,0,3,0,6,2,13,6,28,20,61,56,135,148,308,380,707,950,1654,2340,
%T A339416 3897,5714,9252,13858,22055,33492,52735,80744,126313,194376,302906,
%U A339416 467506,726862,1123830,1744947,2700682,4190016,6488824,10062649,15588714,24168232,37447884
%N A339416 Number of compositions (ordered partitions) of n into an even number of triangular numbers.
%H A339416 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%H A339416 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F A339416 G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k*(k + 1)/2)) + 1 / Sum_{k>=0} x^(k*(k + 1)/2)).
%F A339416 a(n) = (A023361(n) + A106507(n)) / 2.
%F A339416 a(n) = Sum_{k=0..n} A023361(k) * A106507(n-k).
%e A339416 a(9) = 6 because we have [6, 3], [3, 6], [6, 1, 1, 1], [1, 6, 1, 1], [1, 1, 6, 1] and [1, 1, 1, 6].
%p A339416 b:= proc(n, t) option remember; local r, f, g;
%p A339416       if n=0 then t else r, f, g:=$0..2; while f<=n
%p A339416       do r, f, g:= r+b(n-f, 1-t), f+g, g+1 od; r fi
%p A339416     end:
%p A339416 a:= n-> b(n, 1):
%p A339416 seq(a(n), n=0..50);  # _Alois P. Heinz_, Dec 03 2020
%t A339416 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 A339416 Cf. A000217, A023361, A034008, A106507, A339373, A339417.
%K A339416 nonn
%O A339416 0,5
%A A339416 _Ilya Gutkovskiy_, Dec 03 2020