A309536 - OEIS

%I #11 Aug 18 2019 07:11:08

%S 0,1,2,6,14,33,77,174,389,860,1885,4098,8853,19020,40668,86593,183698,

%T 388421,818892,1721884,3611968,7560337,15793474,32932549,68556300,

%U 142495004,295754816,613039248,1269137729,2624393922,5421024773,11186523404,23061994524

%N Total number of triangular numbers in all compositions of n.

%H Alois P. Heinz, <a href="/A309536/b309536.txt">Table of n, a(n) for n = 0..3312</a>

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

%F a(n) ~ c * 2^n * n, where c = 0.1604081401637884665734606925563573585565153844... - _Vaclav Kotesovec_, Aug 18 2019

%e a(4) = 14: (1)(1)(1)(1), 2(1)(1), (1)2(1), (1)(1)2, 22, (3)(1), (1)(3), 4.

%p a:= proc(n) option remember; add(a(n-j)+

%p       `if`(issqr(8*j+1), ceil(2^(n-j-1)), 0), j=1..n)

%p     end:

%p seq(a(n), n=0..33);

%t CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2*x^(1/8)) - 1)*((1 - x)^2/(1 - 2*x)^2), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Aug 18 2019 *)

%Y Cf. A000217, A102291, A263235.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 06 2019