A224677 Number of compositions of n*(n+1)/2 into sums of positive triangular numbers.

1, 1, 2, 7, 40, 351, 4876, 104748, 3487153, 179921982, 14387581923, 1783124902639, 342504341570010, 101962565961894431, 47044167891731682278, 33640402686770010577421, 37282664267078280296013183, 64038780633654058635677191329, 170478465430659361252118580217675

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..102

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

Cf. A023361, A224366, A224679.

Sequence in context: A274279 A319945 A132785 * A064626 A137731 A363004

Adjacent sequences: A224674 A224675 A224676 * A224678 A224679 A224680

Keyword

nonn

Author

Paul D. Hanna, Apr 14 2013

Status

approved