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%I #45 Mar 13 2023 09:11:48
%S 1,1,2,4,7,12,22,43,85,164,308,573,1079,2081,4097,8129,16049,31315,
%T 60402,115806,222416,430791,843987,1670054,3322167,6606936,13078586,
%U 25714238,50230292,97708338,189921842,370216757,725680489,1431888173,2842060970,5662371069
%N Expansion of Sum_{k>=1} (x/(1 - x))^(k*(k+1)/2).
%C Number of compositions of n into a triangular number of parts.
%H Vaclav Kotesovec, <a href="/A280352/a280352.jpg">Plot of a(n) / (2^(n-1)/sqrt(n)) for n = 1..10000</a>
%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F G.f.: Sum_{k>=1} (x/(1-x))^(k*(k+1)/2).
%F 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
%F Conjecture: a(n+1)/a(n) ~ 2. - _Jerzy R Borysowicz_, Jan 14 2023
%F 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
%e a(5) = 7 because we have:
%e [1] [5]
%e [2] [3, 1, 1]
%e [3] [1, 3, 1]
%e [4] [1, 1, 3]
%e [5] [2, 2, 1]
%e [6] [2, 1, 2]
%e [7] [1, 2, 2]
%t nmax = 36; Rest[CoefficientList[Series[Sum[(x/(1 - x))^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]]
%t 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 *)
%o (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
%Y Cf. A000217, A052467, A103198, A280351.
%K nonn,easy
%O 1,3
%A _Ilya Gutkovskiy_, Jan 01 2017
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