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

0, 1, 0, 2, 0, 4, 1, 9, 3, 19, 12, 41, 33, 91, 92, 203, 238, 466, 602, 1080, 1493, 2536, 3661, 6001, 8902, 14278, 21554, 34094, 52013, 81602, 125297, 195582, 301475, 469193, 724881, 1126161, 1742206, 2703888, 4186276, 6493192, 10057553, 15594636, 24161364, 37455851

Offset

0,4

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-1} A023361(k) * A106507(n-k).

Example

a(8) = 3 because we have [6, 1, 1], [1, 6, 1] and [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, 0):
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, A106507, A166444, A339374, A339416.

Sequence in context: A355018 A177256 A199891 * A351558 A226240 A109468

Adjacent sequences: A339414 A339415 A339416 * A339418 A339419 A339420

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 03 2020

Status

approved