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A224677
Number of compositions of n*(n+1)/2 into sums of positive triangular numbers.
7
1, 1, 2, 7, 40, 351, 4876, 104748, 3487153, 179921982, 14387581923, 1783124902639, 342504341570010, 101962565961894431, 47044167891731682278, 33640402686770010577421, 37282664267078280296013183, 64038780633654058635677191329, 170478465430659361252118580217675
OFFSET
0,3
LINKS
FORMULA
a(n) = A023361(n*(n+1)/2), where A023361(n) is the number of compositions of n into positive triangular numbers.
a(n) = [x^(n*(n+1)/2)] 1/(1 - Sum_{k>=1} x^(k*(k+1)/2)).
MAPLE
b:= proc(n) option remember; local i; if n=0 then 1 else 0;
for i while i*(i+1)/2<=n do %+b(n-i*(i+1)/2) od; % fi
end:
a:= n-> b(n*(n+1)/2):
seq(a(n), n=0..20); # Alois P. Heinz, Feb 05 2018
MATHEMATICA
b[n_] := b[n] = If[n==0, 1, Sum[If[IntegerQ[Sqrt[8j+1]], b[n-j], 0], {j, 1, n}]];
a[n_] := b[n(n+1)/2];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 31 2020, after Alois P. Heinz in A023361 *)
PROG
(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)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 14 2013
STATUS
approved