A280352 Expansion of Sum_{k>=1} (x/(1 - x))^(k*(k+1)/2).

1, 1, 2, 4, 7, 12, 22, 43, 85, 164, 308, 573, 1079, 2081, 4097, 8129, 16049, 31315, 60402, 115806, 222416, 430791, 843987, 1670054, 3322167, 6606936, 13078586, 25714238, 50230292, 97708338, 189921842, 370216757, 725680489, 1431888173, 2842060970, 5662371069

Offset: 1

Author

Ilya Gutkovskiy, Jan 01 2017

Keywords

• nonn
• easy
• changed

Comments

Number of compositions of n into a triangular number of parts.

Examples

			a(5) = 7 because we have:
  [1]  [5]
  [2]  [3, 1, 1]
  [3]  [1, 3, 1]
  [4]  [1, 1, 3]
  [5]  [2, 2, 1]
  [6]  [2, 1, 2]
  [7]  [1, 2, 2]
		

Links

• Vaclav Kotesovec, Plot of a(n) / (2^(n-1)/sqrt(n)) for n = 1..10000
• Index entries for sequences related to compositions

Crossrefs

Cf. A000217, A052467, A103198, A280351.

Programs

Mathematica

nmax = 36; Rest[CoefficientList[Series[Sum[(x/(1 - x))^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 40; Rest[CoefficientList[Series[-1 + EllipticTheta[2, 0, Sqrt[x/(1-x)]]/(2*(x/(1-x))^(1/8)), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 01 2017 *)

PARI

a(n) = sum(k=1, (sqrtint(8*n+1)-1)\2, binomial(n-1, k*(k+1)/2-1)) \\ Andrew Howroyd, Jan 14 2023

Formula

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

a(n) = Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} binomial(n-1, k*(k+1)/2-1). - Jerzy R Borysowicz, Dec 26 2022

Conjecture: a(n+1)/a(n) ~ 2. - Jerzy R Borysowicz, Jan 14 2023

Conjecture: abs(b(n)-1) < 0.015, where b(n) = a(n)*sqrt(n)/2^(n-1), for n > 781; b(n) does not have a limit. - Jerzy R Borysowicz, Feb 17 2023