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%I #16 Oct 31 2020 12:41:00
%S 1,1,2,7,40,351,4876,104748,3487153,179921982,14387581923,
%T 1783124902639,342504341570010,101962565961894431,
%U 47044167891731682278,33640402686770010577421,37282664267078280296013183,64038780633654058635677191329,170478465430659361252118580217675
%N Number of compositions of n*(n+1)/2 into sums of positive triangular numbers.
%H Alois P. Heinz, <a href="/A224677/b224677.txt">Table of n, a(n) for n = 0..102</a>
%F a(n) = A023361(n*(n+1)/2), where A023361(n) is the number of compositions of n into positive triangular numbers.
%F a(n) = [x^(n*(n+1)/2)] 1/(1 - Sum_{k>=1} x^(k*(k+1)/2)).
%p b:= proc(n) option remember; local i; if n=0 then 1 else 0;
%p for i while i*(i+1)/2<=n do %+b(n-i*(i+1)/2) od; % fi
%p end:
%p a:= n-> b(n*(n+1)/2):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Feb 05 2018
%t b[n_] := b[n] = If[n==0, 1, Sum[If[IntegerQ[Sqrt[8j+1]], b[n-j], 0], {j, 1, n}]];
%t a[n_] := b[n(n+1)/2];
%t a /@ Range[0, 20] (* _Jean-François Alcover_, Oct 31 2020, after _Alois P. Heinz_ in A023361 *)
%o (PARI) {a(n)=polcoeff(1/(1-sum(r=1,n+1, x^(r*(r+1)/2)+x*O(x^(n*(n+1)/2)))), n*(n+1)/2)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A023361, A224366, A224679.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 14 2013
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