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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).

%I A116679 #11 Jun 09 2019 11:53:59
%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,
%T A116679 6,1,3,10,8,1,4,11,10,2,5,13,11,3,5,15,14,4,5,18,18,5,6,20,21,7,7,23,
%U A116679 24,9,1,8,26,29,12,1,8,30,36,14,1,9,34,41,18,2,11,38,47,23,3,12,43,55,28,4
%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+4*n))/2) terms.
%C A116679 Row sums yield A000009.
%C A116679 T(n,0) = A000700(n), T(n,1) = A096911(n) for n >= 1.
%C A116679 Sum_{k>=0} k*T(n,k) = A116680(n).
%H A116679 G. C. Greubel, <a href="/A116679/b116679.txt">Rows n = 0..100 of triangle, flattened</a>
%F A116679 G.f.: Product_{j>=1} (1+x^(2*j-1))*(1+t*x^(2*j)).
%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
%t A116679 With[{m=25}, CoefficientList[CoefficientList[Series[Product[(1+x^(2*j- 1))*(1+t*x^(2*j)), {j,1,m+2}], {x,0,m}, {t,0,m}], x], t]]//Flatten (* _G. C. Greubel_, Jun 07 2019 *)
%Y A116679 Cf. A000009, A000700, A096911, A116680.
%K A116679 nonn,tabf
%O A116679 0,10
%A A116679 _Emeric Deutsch_, Feb 22 2006