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A276426 Triangle read by rows: T(n,k) is the number of integer partitions of n having k distinct odd parts (n>=0).

%I A276426 #16 Feb 07 2017 11:50:10
%S A276426 1,0,1,1,1,0,3,2,2,1,0,6,1,3,5,3,0,11,4,5,8,9,0,20,9,1,7,15,19,1,0,32,
%T A276426 21,3,11,24,38,4,0,51,41,9,15,39,69,12,0,80,73,23,22,58,123,27,1,0,
%U A276426 119,128,49,1,30,90,202,60,3,0,175,213,98,4,42,130,328,118,9
%N A276426 Triangle read by rows: T(n,k) is the number of integer partitions of n having k distinct odd parts (n>=0).
%C A276426 Sum of entries in row n = A000041(n).
%C A276426 T(2n,0) = A000041(n); T(2n+1,0) = 0.
%C A276426 Sum(k*T(n,k), k>=0) = A024786(n+1).
%H A276426 Alois P. Heinz, <a href="/A276426/b276426.txt">Rows n = 0..1000, flattened</a>
%F A276426 G.f.:  G(t,x) = Product_{j>=1} ((1-(1-t)*x^{2*j-1})/(1-x^j)).
%e A276426 T(4,0) = 2 because we have [4], [2,2];
%e A276426 T(4,1) = 2 because we have [1,1,2], [1,1,1,1];
%e A276426 T(4,2) = 1 because we have [1,3];
%e A276426 Triangle starts:
%e A276426 1;
%e A276426 0,1;
%e A276426 1,1;
%e A276426 0,3;
%e A276426 2,2,1.
%p A276426 G := product((1-x^(2*j-1)+t*x^(2*j-1))/(1-x^j), j = 1 .. 100): Gser := simplify(series(G, x = 0, 32)); for n from 0 to 27 do P[n] := sort(coeff(Gser, x, n)) end do: for n from 0 to 27 do seq(coeff(P[n], t, i), i = 0 .. degree(P[n])) end do; # yields sequence in triangular form
%p A276426 # second Maple program:
%p A276426 b:= proc(n, i) option remember; expand(
%p A276426       `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
%p A276426       `if`(j>0 and i::odd, x, 1), j=0..n/i))))
%p A276426     end:
%p A276426 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A276426 seq(T(n), n=0..25);  # _Alois P. Heinz_, Sep 20 2016
%t A276426 b[n_, i_] := b[n, i] = Expand[If[n==0, 1, If[i<1, 0, Sum[b[n-i*j, i-1]*If[j > 0 && OddQ[i], x, 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 25}] // Flatten (* _Jean-François Alcover_, Feb 07 2017, after _Alois P. Heinz_ *)
%Y A276426 Cf. A000041, A024786, A264052.
%K A276426 nonn,tabf
%O A276426 0,7
%A A276426 _Emeric Deutsch_, Sep 19 2016