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A265255 Triangle read by rows: T(n,k) is the number of partitions of n having k odd singletons (n, k >=0).

%I A265255 #18 Dec 10 2016 06:24:05
%S A265255 1,0,1,2,1,2,4,0,1,2,5,8,1,2,4,11,14,3,5,9,20,0,1,24,8,10,16,37,1,2,
%T A265255 41,15,21,28,65,3,5,66,30,39,49,108,9,10,104,57,69,0,1,80,178,19,20,
%U A265255 163,99,120,1,2,128,286,39,37,248,170,201,3,5,203,448,73,68,372,284,327
%N A265255 Triangle read by rows: T(n,k) is the number of partitions of n having k odd singletons (n, k >=0).
%C A265255 Sum of entries in row n is A000041(n).
%C A265255 T(n,0) = A265256(n).
%C A265255 Sum_{k>=0} k*T(n,k) = A265257(n).
%H A265255 Alois P. Heinz, <a href="/A265255/b265255.txt">Rows n = 0..1000, flattened</a>
%F A265255 G.f.: G(t,x) = Product_{j>=1} ((1 -x^(2j-1))(1+tx^{2j-1}) + x^(4j-2))/ (1-x^j).
%e A265255 T(6,2) = 2 because each of the partitions [1,2,3], [1,5] of n = 6 has 2 odd singleton, while the other partitions, namely [1,1,1,1,1,1], [1,1,1,1,2], [1,1,2,2], [2,2,2], [1,1,1,3], [3,3], [1,1,4], [2,4], [6], have 0, 0, 0, 0, 1, 0, 0, 0, 0  odd singletons.
%e A265255 Triangle starts:
%e A265255 1;
%e A265255 0, 1;
%e A265255 2;
%e A265255 1, 2;
%e A265255 4, 0, 1;
%e A265255 2, 5;
%e A265255 8, 1, 2.
%p A265255 g := mul(((1-x^(2*j-1))*(1+t*x^(2*j-1))+x^(4*j-2))/(1-x^j), j = 1 .. 80): gser := simplify(series(g, x = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, q), q = 0 .. degree(P[n])) end do; # yields sequence in triangular form
%p A265255 # second Maple program:
%p A265255 b:= proc(n, i) option remember; expand(
%p A265255       `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
%p A265255       `if`(j=1 and i::odd, x, 1), j=0..n/i))))
%p A265255     end:
%p A265255 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A265255 seq(T(n), n=0..30);  # _Alois P. Heinz_, Jan 01 2016
%t A265255 b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*If[j == 1 && 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, 30}] // Flatten (* _Jean-François Alcover_, Dec 10 2016, after _Alois P. Heinz_ *)
%Y A265255 Cf. A000041, A265253, A265256, A265257.
%K A265255 nonn,tabf,look
%O A265255 0,4
%A A265255 _Emeric Deutsch_, Jan 01 2016