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A224679
Number of compositions of n^2 into sums of positive triangular numbers.
4
1, 1, 3, 25, 546, 28136, 3487153, 1038115443, 742336894991, 1275079195875471, 5260826667789867957, 52137661179700350278531, 1241165848412448464485760897, 70972288312605764017275784402928, 9748291749334923037419108242002717050
OFFSET
0,3
LINKS
FORMULA
a(n) = A023361(n^2), where A023361(n) = number of compositions of n into positive triangular numbers.
a(n) = [x^(n^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^2):
seq(a(n), n=0..20); # Alois P. Heinz, Feb 05 2018
MATHEMATICA
b[n_] := b[n] = Module[{i, j = If[n == 0, 1, 0]}, For[i = 1, i(i+1)/2 <= n, i++, j += b[n-i(i+1)/2]]; j];
a[n_] := b[n^2];
a /@ Range[0, 20] (* Jean-François Alcover, Nov 04 2020, after Alois P. Heinz *)
PROG
(PARI) {a(n)=polcoeff(1/(1-sum(r=1, n+1, x^(r*(r+1)/2)+x*O(x^(n^2)))), n^2)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 14 2013
STATUS
approved