%I A116679
%S A116679 1,1,0,1,1,1,1,1,1,2,1,2,1,1,3,1,2,3,1,2,4,2,2,5,3,2,6,4,3,7,4,1,3,8,6,
%T A116679 1,3,10,8,1,4,11,10,2,5,13,11,3,5,15,14,4,5,18,18,5,6,20,21,7,7,23,24,
               9,
%U A116679 1,8,26,29,12,1,8,30,36,14,1,9,34,41,18,2,11,38,47,23,3,12,43,55,28,4
%N A116679 Triangle read by rows: T(n,k) is the number of partitions of n into distinct 
               part and having exactly k even parts (n>=0, k>=0).
%C A116679 Row n contains floor((1+sqrt(1+4n))/2) terms. Row sums yield A000009. 
               T(n,0)=A000700(n). T(n,1)=A096911(n) (n>=1). Sum(k*T(n,k), k>=0)=A116680(n).
%F A116679 G.f.=product((1+x^(2j-1))(1+tx^(2j)), j=1..infinity).
%e A116679 T(9,2)=2 because we have [6,2,1] and [4,3,2].
%e A116679 Triangle starts:
%e A116679 1;
%e A116679 1;
%e A116679 0,1;
%e A116679 1,1;
%e A116679 1,1;
%e A116679 1,2;
%e A116679 1,2,1;
%e A116679 1,3,1;
%p A116679 g:=product((1+x^(2*j-1))*(1+t*x^(2*j)),j=1..25): gser:=simplify(series(g,
               x=0,38)): P[0]:=1: for n from 1 to 27 do P[n]:=sort(coeff(gser,x^n)) 
               od: for n from 0 to 27 do seq(coeff(P[n],t,j),j=0..floor((sqrt(1+4*n)-1)/
               2)) od; # yields sequence in triangular form
%Y A116679 Cf. A000009, A000700, A096911, A116680.
%Y A116679 Sequence in context: A129479 A075104 A008289 this_sequence A146290 A135539 
               A129264
%Y A116679 Adjacent sequences: A116676 A116677 A116678 this_sequence A116680 A116681 
               A116682
%K A116679 nonn,tabf
%O A116679 0,10
%A A116679 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 22 2006