cp's OEIS Frontend

A113686 Triangular array T(n,k)=number of partitions of n in which sum of even parts is k, for k=0,1,...n; n>=0.

%I A113686 #5 Mar 30 2012 18:57:06
%S A113686 1,1,0,1,0,1,2,0,1,0,2,0,1,0,2,3,0,2,0,2,0,4,0,2,0,2,0,3,5,0,3,0,4,0,
%T A113686 3,0,6,0,4,0,4,0,3,0,5,8,0,5,0,6,0,6,0,5,0,10,0,6,0,8,0,6,0,5,0,7,12,
%U A113686 0,8,0,10,0,9,0,10,0,7,0,15,0,10,0,12,0,12,0,10,0,7,0,11,18,0,12,0,16,0,15
%N A113686 Triangular array T(n,k)=number of partitions of n in which sum of even parts is k, for k=0,1,...n; n>=0.
%C A113686 (Sum over row n) = A000041(n) = number of partitions of n. Reversal of this array is array in A113685, except for row 0.
%C A113686 Sum(k*T(n,k),k=0..n)=A066966(n). - _Emeric Deutsch_, Feb 17 2006
%F A113686 G:=1/product((1-x^(2j-1))(1-t^(2j)x^(2j)), j=1..infinity). - _Emeric Deutsch_, Feb 17 2006
%e A113686 First 5 rows:
%e A113686 1
%e A113686 1 0
%e A113686 1 0 1
%e A113686 2 0 1 0
%e A113686 2 0 1 0 2
%e A113686 3 0 2 0 2 0.
%e A113686 The partitions of 5 are
%e A113686 5, 1+4, 2+3, 1+1+3, 1+2+2, 1+1+1+2, 1+1+1+1+1;
%e A113686 sums of even parts are 0,4,2,0,4,2, respectively,
%e A113686 so that the numbers of 0's, 1's, 2s, 3s, 4s, 5s
%e A113686 are 0,3,0,2,0,2,0, which is row 5 of the array.
%p A113686 g:=1/product((1-x^(2*j-1))*(1-t^(2*j)*x^(2*j)),j=1..20): gser:=simplify(series(g,x=0,20)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(gser,x^n) od: for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form - _Emeric Deutsch_, Feb 17 2006
%Y A113686 Cf. A000041, A113685.
%Y A113686 Cf. A066966.
%K A113686 nonn,tabl
%O A113686 0,7
%A A113686 _Clark Kimberling_, Nov 05 2005
%E A113686 More terms from _Emeric Deutsch_, Feb 17 2006