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

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, 10, 1, 43, 43, 14, 1, 54, 59, 20, 2, 70, 76, 27, 3, 89, 99, 38, 5, 111, 129, 50, 7, 140, 165, 69, 11, 174, 211, 90, 15, 216, 270, 119, 21, 1, 268, 339, 155, 29, 1, 328, 429, 203, 40, 2

Offset: 0

Author

Emeric Deutsch, Dec 31 2015

Keywords

• nonn
• tabf

Comments

T(n,0) = A265254(n).

Sum(k*T(n,k), k>=0) = A024788(n+2).

Examples

			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.
Triangle starts:
  1;
  1;
  1, 1;
  2, 1;
  3, 2;
  4, 3;
  6, 4, 1.
		

Links

• Alois P. Heinz, Rows n = 0..1000, flattened

Crossrefs

Cf. A024788, A265254.

Programs

Maple

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
# second Maple program:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
      `if`(j=1 and i::even, x, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..30);  # Alois P. Heinz, Jan 01 2016

Mathematica

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 *)

Formula

G.f.: G(t,x) = Product_{j>=1}((1-x^{2j})(1+tx^{2j}) + x^{4j})/(1-x^j).