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A264399 Triangle read by rows: T(n,k) is the number of partitions of n having k parts with even multiplicities. 4
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A227634 A064885 A029618 * A240225 A283893 A112427

Adjacent sequences:  A264396 A264397 A264398 * A264400 A264401 A264402

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Nov 21 2015

STATUS

approved

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Last modified July 24 02:56 EDT 2019. Contains 325290 sequences. (Running on oeis4.)