cp's OEIS Frontend

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

%I A224679 #8 Nov 04 2020 05:55:20
%S A224679 1,1,3,25,546,28136,3487153,1038115443,742336894991,1275079195875471,
%T A224679 5260826667789867957,52137661179700350278531,
%U A224679 1241165848412448464485760897,70972288312605764017275784402928,9748291749334923037419108242002717050
%N A224679 Number of compositions of n^2 into sums of positive triangular numbers.
%H A224679 Alois P. Heinz, <a href="/A224679/b224679.txt">Table of n, a(n) for n = 0..72</a>
%F A224679 a(n) = A023361(n^2), where A023361(n) = number of compositions of n into positive triangular numbers.
%F A224679 a(n) = [x^(n^2)] 1/(1 - Sum_{k>=1} x^(k*(k+1)/2)).
%p A224679 b:= proc(n) option remember; local i; if n=0 then 1 else 0;
%p A224679       for i while i*(i+1)/2<=n do %+b(n-i*(i+1)/2) od; %  fi
%p A224679     end:
%p A224679 a:= n-> b(n^2):
%p A224679 seq(a(n), n=0..20);  # _Alois P. Heinz_, Feb 05 2018
%t A224679 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];
%t A224679 a[n_] := b[n^2];
%t A224679 a /@ Range[0, 20] (* _Jean-François Alcover_, Nov 04 2020, after _Alois P. Heinz_ *)
%o A224679 (PARI) {a(n)=polcoeff(1/(1-sum(r=1,n+1, x^(r*(r+1)/2)+x*O(x^(n^2)))), n^2)}
%o A224679 for(n=0, 20, print1(a(n), ", "))
%Y A224679 Cf. A023361, A224366, A224677.
%K A224679 nonn
%O A224679 0,3
%A A224679 _Paul D. Hanna_, Apr 14 2013