A264398 Triangle read by rows: T(n,k) is the number of partitions of n having k parts with odd multiplicities.

1, 0, 1, 1, 1, 0, 2, 1, 2, 2, 1, 0, 4, 3, 3, 4, 3, 1, 0, 7, 7, 1, 5, 7, 7, 3, 0, 12, 14, 4, 7, 12, 14, 8, 1, 0, 19, 26, 10, 1, 11, 19, 26, 18, 3, 0, 30, 45, 22, 4, 15, 30, 45, 36, 9, 0, 45, 75, 44, 11, 1, 22, 45, 75, 67, 21, 1, 0, 67, 120, 81, 26, 3, 30, 67, 120, 119, 45, 4

Offset: 0

Author

Emeric Deutsch, Nov 21 2015

Keywords

• nonn
• look
• tabf

Comments

T(n,0) = A035363(n).

Row sums give A000041. - Omar E. Pol, Nov 21 2015

Examples

			T(6,1) = 4 because we have [6*], [4*,1,1],[2*,2,2], and [2*,1,1,1,1] (parts with odd multiplicities are marked).
Triangle starts:
  1;
  0, 1;
  1, 1;
  0, 2, 1;
  2, 2, 1;
  0, 4, 3;
  3, 4, 3, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..1000, flattened
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 30

Crossrefs

Cf. A000041, A035363, A209423, A264399, A264400.

Programs

Maple

g := product((1+t*x^j)/(1-x^(2*j)), j = 1 .. 100): gser := simplify(series(g, x = 0, 30)): for n from 0 to 28 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 23 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(expand(`if`(j::odd, x, 1)*b(n-i*j, i-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..20);  # Alois P. Heinz, Nov 25 2015

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Expand[If[OddQ[j], x, 1]* b[n-i*j, i-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, 20}] // Flatten (* Jean-François Alcover, Feb 18 2016, after Alois P. Heinz *)

PARI

T(n) = { Vec(prod(k=1, n, (1 + y*x^k)/(1 - x^(2*k)) + O(x*x^n))) }
{ my(t=T(10)); for(n=1, #t, print(Vecrev(t[n]))); } \\ Andrew Howroyd, Dec 22 2017

Formula

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

Sum_{k>0} k * T(n,k) = A209423(n). - Alois P. Heinz, Aug 05 2020

G.f.: A(x,y)*B(x^2) where A(x),B(x) are the o.g.f.'s for A008289 and A000041. (See Flajolet, Sedgewick link.) - Geoffrey Critzer, Aug 07 2022