A224679 Number of compositions of n^2 into sums of positive triangular numbers.

1, 1, 3, 25, 546, 28136, 3487153, 1038115443, 742336894991, 1275079195875471, 5260826667789867957, 52137661179700350278531, 1241165848412448464485760897, 70972288312605764017275784402928, 9748291749334923037419108242002717050

Offset: 0

Author

Paul D. Hanna, Apr 14 2013

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..72

Crossrefs

Cf. A023361, A224366, A224677.

Programs

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 *)

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), ", "))

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)).