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A265253 Triangle read by rows: T(n,k) is the number of partitions of n having k even singletons (n,k>=0).

%I A265253 #17 Jan 20 2016 09:11:07
%S A265253 1,1,1,1,2,1,3,2,4,3,6,4,1,8,6,1,11,9,2,15,12,3,19,18,5,25,24,7,34,32,
%T A265253 10,1,43,43,14,1,54,59,20,2,70,76,27,3,89,99,38,5,111,129,50,7,140,
%U A265253 165,69,11,174,211,90,15,216,270,119,21,1,268,339,155,29,1,328,429,203,40,2
%N A265253 Triangle read by rows: T(n,k) is the number of partitions of n having k even singletons (n,k>=0).
%C A265253 T(n,0) = A265254(n).
%C A265253 Sum(k*T(n,k), k>=0) = A024788(n+2).
%H A265253 Alois P. Heinz, <a href="/A265253/b265253.txt">Rows n = 0..1000, flattened</a>
%F A265253 G.f.: G(t,x) = Product_{j>=1}((1-x^{2j})(1+tx^{2j}) + x^{4j})/(1-x^j).
%e A265253 T(6,1) = 4 because each of the partitions [1,1,1,1,2], [1,2,3], [1,1,4], [6] of n = 6 has 1 even singleton, while the other partitions, namely [1,1,1,1,1,1], [1,1,2,2], [2,2,2], [1,1,1,3], [3,3], [2,4], [1,5], have 0, 0, 0 ,0, 0, 2, 0 even singletons.
%e A265253 Triangle starts:
%e A265253   1;
%e A265253   1;
%e A265253   1, 1;
%e A265253   2, 1;
%e A265253   3, 2;
%e A265253   4, 3;
%e A265253   6, 4, 1.
%p A265253 g := mul(((1-x^(2*j))*(1+t*x^(2*j))+x^(4*j))/(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 A265253 # second Maple program:
%p A265253 b:= proc(n, i) option remember; expand(
%p A265253       `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
%p A265253       `if`(j=1 and i::even, x, 1), j=0..n/i))))
%p A265253     end:
%p A265253 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A265253 seq(T(n), n=0..30);  # _Alois P. Heinz_, Jan 01 2016
%t A265253 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 && EvenQ[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_, Jan 20 2016, after _Alois P. Heinz_ *)
%Y A265253 Cf. A024788, A265254.
%K A265253 nonn,tabf
%O A265253 0,5
%A A265253 _Emeric Deutsch_, Dec 31 2015