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

1, 0, 1, 0, 3, 0, 6, 2, 13, 6, 28, 20, 61, 56, 135, 148, 308, 380, 707, 950, 1654, 2340, 3897, 5714, 9252, 13858, 22055, 33492, 52735, 80744, 126313, 194376, 302906, 467506, 726862, 1123830, 1744947, 2700682, 4190016, 6488824, 10062649, 15588714, 24168232, 37447884

Offset: 0

Author

Ilya Gutkovskiy, Dec 03 2020

Keywords

• nonn

Examples

			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].
		

Links

• Index entries for sequences related to compositions
• Index to sequences related to polygonal numbers

Crossrefs

Cf. A000217, A023361, A034008, A106507, A339373, A339417.

Programs

Maple

b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+g, g+1 od; r fi
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 03 2020

Mathematica

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]

Formula

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k*(k + 1)/2)) + 1 / Sum_{k>=0} x^(k*(k + 1)/2)).

a(n) = (A023361(n) + A106507(n)) / 2.

a(n) = Sum_{k=0..n} A023361(k) * A106507(n-k).