A385604 Number of compositions of n such that the odd parts are weakly increasing.

1, 1, 2, 4, 7, 14, 25, 48, 86, 162, 292, 541, 978, 1794, 3247, 5919, 10712, 19451, 35184, 63729, 115199, 208327, 376333, 679842, 1227403, 2215695, 3998408, 7214274, 13014001, 23472678, 42331028, 76330880, 137627168, 248122171, 447301570, 806312371, 1453405651

Offset

0,3

Links

Table of n, a(n) for n=0..36.

Formula

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

Example

a(5) = 14 counts all compositions of n = 5 except (1,3,1) and (3,1,1) since the odd parts are not weakly increasing.
The composition of n = 13 (2,1,1,4,2,3) has odd parts (1,1,3), so it is counted under a(13) = 1794.

Prog

(PARI) A_x(N) = {my(x='x+O('x^(N+1))); Vec((1-x^2)/(1-2*x^2)/prod(i=0, N, 1-x^(2*i+1)*(1-x^2)/(1-2*x^2)))}

Crossrefs

Cf. A000009, A011782, A032021, A098123, A289249.

Sequence in context: A218576 A054169 A287185 * A065491 A072810 A167606

Adjacent sequences: A385601 A385602 A385603 * A385605 A385606 A385607

Keyword

nonn,easy

Author

John Tyler Rascoe, Aug 02 2025

Status

approved