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Triangle read by rows: T(n,k) is the number of integer partitions of n having k distinct odd parts (n>=0).
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%I #16 Feb 07 2017 11:50:10

%S 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 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 119,128,49,1,30,90,202,60,3,0,175,213,98,4,42,130,328,118,9

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

%C Sum of entries in row n = A000041(n).

%C T(2n,0) = A000041(n); T(2n+1,0) = 0.

%C Sum(k*T(n,k), k>=0) = A024786(n+1).

%H Alois P. Heinz, <a href="/A276426/b276426.txt">Rows n = 0..1000, flattened</a>

%F G.f.: G(t,x) = Product_{j>=1} ((1-(1-t)*x^{2*j-1})/(1-x^j)).

%e T(4,0) = 2 because we have [4], [2,2];

%e T(4,1) = 2 because we have [1,1,2], [1,1,1,1];

%e T(4,2) = 1 because we have [1,3];

%e Triangle starts:

%e 1;

%e 0,1;

%e 1,1;

%e 0,3;

%e 2,2,1.

%p 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 # second Maple program:

%p b:= proc(n, i) option remember; expand(

%p `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*

%p `if`(j>0 and i::odd, x, 1), j=0..n/i))))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):

%p seq(T(n), n=0..25); # _Alois P. Heinz_, Sep 20 2016

%t 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 Cf. A000041, A024786, A264052.

%K nonn,tabf

%O 0,7

%A _Emeric Deutsch_, Sep 19 2016