A264399 Triangle read by rows: T(n,k) is the number of partitions of n having k parts with even multiplicities.

1, 1, 1, 1, 3, 2, 3, 5, 2, 6, 4, 1, 9, 6, 9, 11, 2, 16, 13, 1, 20, 15, 7, 25, 28, 3, 32, 33, 11, 1, 40, 52, 9, 54, 55, 24, 2, 69, 82, 25, 84, 101, 40, 6, 101, 148, 46, 2, 136, 163, 73, 13, 156, 239, 89, 6, 202, 274, 127, 23, 1, 244, 364, 170, 14, 306, 437, 211, 46, 2

Offset

0,5

Comments

T(n,0) = A055922(n).

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

Links

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

Formula

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

Example

T(6,1) = 4 because we have [4,1*,1], [3*,3], [2,1*,1,1,1], and [1*,1,1,1,1,1] (parts with even multiplicities are marked).
Triangle starts:
  1;
  1;
  1, 1;
  3;
  2, 3;
  5, 2;
  6, 4, 1;
  ...

Maple

g := product(1+x^j/(1-x^(2*j))+t*x^(2*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 28 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>0 and j::even, 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..30);  # Alois P. Heinz, Nov 25 2015

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Expand[If[j>0 && EvenQ[ 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, 30}] // Flatten (* Jean-François Alcover, Dec 25 2015, after Alois P. Heinz *)

Prog

(PARI)
T(n) = { Vec(prod(k=1, n, (1+x^k-x^(2*k)+y*x^(2*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

Crossrefs

Cf. A000041, A055922, A264398, A264400.

Sequence in context: A064885 A029618 A330656 * A240225 A283893 A112427

Adjacent sequences: A264396 A264397 A264398 * A264400 A264401 A264402

Keyword

nonn,look,tabf

Author

Emeric Deutsch, Nov 21 2015

Status

approved