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

0,3

Links

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

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

Cf. A023361, A224366, A224677.

Sequence in context: A243440 A306783 A003024 * A213599 A179473 A248417

Adjacent sequences: A224676 A224677 A224678 * A224680 A224681 A224682

Keyword

nonn

Author

Paul D. Hanna, Apr 14 2013

Status

approved