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%I #10 May 13 2015 05:09:52
%S 1,0,1,1,0,0,2,1,0,1,0,3,0,2,0,2,0,5,0,2,0,4,0,7,0,1,3,0,7,0,0,10,0,2,
%T 4,0,11,0,0,14,0,4,5,0,17,0,0,19,0,8,6,0,25,0,1,0,25,0,13,0,8,0,36,0,
%U 2,0,33,0,21,0,10,0,50,0,4,0,43,0,33,0,12,0,69,0,8,0,55,0,49,0,15,0,93,0,14
%N Triangle read by rows: T(n,k) is the number of partitions of n into distinct part and having exactly k odd parts (n>=0, k>=0).
%C Row n contains 1+floor(sqrt(n)) terms (at the end of certain rows there is an extra 0). Row sums yield A000009. T(n,0) = A035457(n) (n>=1); T(2n,0) = A000009(n), T(2n-1,0)=0. T(2n,1)=0, T(2n-1,1) = A036469(n). Sum(k*T(n,k), k>=0) = A116676(n).
%H Alois P. Heinz, <a href="/A116675/b116675.txt">Rows n = 0..750, flattened</a>
%F G.f.: product((1+tx^(2j-1))(1+x^(2j)), j=1..infinity).
%e T(8,2) = 4 because we have [7,1], [5,3], [5,2,1] and [4,3,1] ([8] and [6,2] do not qualify).
%e Triangle starts:
%e 1;
%e 0, 1;
%e 1, 0;
%e 0, 2;
%e 1, 0, 1;
%e 0, 3, 0;
%p g:=product((1+t*x^(2*j-1))*(1+x^(2*j)),j=1..25): gser:=simplify(series(g,x=0,38)): P[0]:=1: for n from 1 to 26 do P[n]:=sort(coeff(gser,x^n)) od: for n from 0 to 26 do seq(coeff(P[n],t,j),j=0..floor(sqrt(n))) od; # yields sequence in triangular form
%p # second Maple program:
%p b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)->
%p x+y, b(n, i-1), `if`(i>n, [], [`if`(irem(i,2)=0, [][], 0),
%p b(n-i, i-1)[]]), 0)))
%p end:
%p T:= proc(n) local l; l:= b(n, n); l[], 0$(1+floor(sqrt(n))-nops(l)) end:
%p seq (T(n), n=0..30); # _Alois P. Heinz_, Nov 21 2012
%t rows = 25; coes = CoefficientList[Product[(1+t*x^(2j-1))(1+x^(2j)), {j, 1, rows}], {x, t}][[1 ;; rows]]; MapIndexed[Take[#1, Floor[Sqrt[#2[[1]]-1]]+1]&, coes] // Flatten (* _Jean-François Alcover_, May 13 2015 *)
%Y Cf. A000009, A035457, A036469, A116676.
%K nonn,tabf
%O 0,7
%A _Emeric Deutsch_, Feb 22 2006
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