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

0,5

Links

Table of n, a(n) for n=0..43.

Index entries for sequences related to compositions

Index to sequences related to polygonal numbers

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).

Example

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].

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]

Crossrefs

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

Sequence in context: A011079 A334874 A343903 * A226535 A005928 A113062

Adjacent sequences: A339413 A339414 A339415 * A339417 A339418 A339419

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 03 2020

Status

approved