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Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of its odd singletons is k (0<=k<=n). A singleton in a partition is a part that occurs exactly once.
4

%I #18 Oct 04 2016 06:00:27

%S 1,0,1,2,0,0,1,1,0,1,4,0,0,0,1,2,2,0,2,0,1,8,0,0,1,1,0,1,4,4,0,4,0,2,

%T 0,1,14,0,0,2,2,1,1,0,2,9,6,0,7,0,4,0,2,0,2,24,1,0,4,3,2,2,1,3,0,2,16,

%U 10,0,12,0,8,0,4,1,3,0,2,41,1,0,7,5,4,4,2,6,1,3,0,3,28,16,0,20,0,14,0,8,2,6,1,3,0,3

%N Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of its odd singletons is k (0<=k<=n). A singleton in a partition is a part that occurs exactly once.

%C T(n,0) = A265256(n).

%C T(n,n) = A000700(n).

%C Sum(k*T(n,k), k>=0) = A276423(n).

%C Sum(T(n,k), k>=0) = A000041(n).

%H Alois P. Heinz, <a href="/A276422/b276422.txt">Rows n = 0..200, flattened</a>

%F G.f.: G(t,x) = Product(((1-x^{2j-1})(1+t^{2j-1}x^{2j-1}) + x^{4j-2})/(1-x^j), j=1..infinity).

%e Row 4 is 4, 0, 0, 0, 1 because in the partitions [1,1,1,1], [1,1,2], [2,2], [1,3], [4] the sums of the odd singletons are 0, 0, 0, 4, 0, respectively.

%e Row 5 is 2, 2, 0, 2, 0, 1 because in the partitions [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5] the sums of the odd singletons are 0, 0, 1, 3, 3, 1, 5, respectively.

%e Triangle starts:

%e 1;

%e 0,1;

%e 2,0,0;

%e 1,1,0,1;

%e 4,0,0,0,1;

%e 2,2,0,2,0,1.

%p g := Product(((1-x^(2*j-1))*(1+t^(2*j-1)*x^(2*j-1))+x^(4*j-2))/(1-x^j), j = 1 .. 100): gser := simplify(series(g, x = 0, 23)): for n from 0 to 20 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 20 do seq(coeff(P[n], t, i), i = 0 .. 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=1 and i::odd, x^i, 1), j=0..n/i))))

%p end:

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

%p seq(T(n), n=0..14); # _Alois P. Heinz_, Sep 14 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 == 1 && OddQ[i], x^i, 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]]; Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Oct 04 2016, after _Alois P. Heinz_ *)

%Y Cf. A000041, A000700, A265256, A276423, A276424, A276425.

%K nonn,tabl

%O 0,4

%A _Emeric Deutsch_, Sep 14 2016