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%I #6 Apr 28 2018 17:13:50
%S 1,2,3,6,9,14,25,38,57,84,141,206,307,440,633,984,1419,2036,2887,4064,
%T 5619,8370,11667,16424,22717,31478,42783,58386,82701,113162,155029,
%U 210770,285645,383688,514497,682922,940327,1256300,1687365,2245692,2997183,3955448,5233315,6854588,8978175,11998806
%N Expansion of (1/(1 - x))*Sum_{k>=0} k!*x^(k*(k+1)/2)/Product_{j=1..k} (1 - x^j).
%C Partial sums of A032020.
%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%p T:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
%p `if`(k=0, `if`(n=0, 1, 0), T(n-k, k) +k*T(n-k, k-1)))
%p end:
%p b:= n-> add(T(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
%p a:= proc(n) option remember;
%p `if`(n<0, 0, b(n)+a(n-1))
%p end:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Apr 28 2018
%t nmax = 45; CoefficientList[Series[1/(1 - x) Sum[k! x^(k (k + 1)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
%Y Cf. A000079, A032020, A036469.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Apr 28 2018
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