A264052 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A259361(n)) is the number of integer partitions of n having k distinct parts occurring at least twice.

1, 1, 1, 1, 2, 1, 2, 3, 3, 4, 4, 6, 1, 5, 9, 1, 6, 13, 3, 8, 18, 4, 10, 23, 9, 12, 32, 12, 15, 42, 19, 1, 18, 55, 27, 1, 22, 69, 41, 3, 27, 89, 56, 4, 32, 112, 78, 9, 38, 141, 106, 12, 46, 175, 141, 23, 54, 217, 188, 31, 64, 266, 247, 49, 1, 76, 326, 321, 68, 1

Offset

0,5

Comments

Row sums give A000041.

T(n,k) is also the number of integer partitions of n having k parts from which one can subtract 2 and still get an integer partition (mapping a partition to its conjugate sends one statistic to the other).

T(n,k) is also the number of integer partitions of n having k distinct even parts. Example: T(6,2)= 1, counting the partition [2,4]. - Emeric Deutsch, Sep 19 2016

Links

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

FindStat - Combinatorial Statistic Finder, The number of distinct parts of a partition that occur at least twice, The number of parts from which one can subtract 2 and still get an integer partition.

V. V. Tewari, Kronecker Coefficients For Some Near-Rectangular Partitions, arXiv:1403.5327 [math.CO], 2014, MathSciNet:3320625.

Formula

From Emeric Deutsch, Nov 12 2015: (Start)

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

T(n,0) = A000009(n).

T(n,1) = A090867(n).

Sum_{k>=0} k*T(n,k) = A024786(n).

(End)

Example

Triangle begins:
   1,
   1,
   1,  1,
   2,  1,
   2,  3,
   3,  4,
   4,  6, 1,
   5,  9, 1,
   6, 13, 3,
   8, 18, 4,
  10, 23, 9,
  ...
T(6,2)= 1; namely [1,1,2,2]. - Emeric Deutsch, Sep 19 2016

Maple

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, 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..25);  # Alois P. Heinz, Nov 02 2015
# second Maple program:
g := product((1-(1-t)*x^(2*j))/(1-x^j), j = 1 .. 80): gser := simplify(series(g, x = 0, 25)): for n from 0 to 23 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 23 do seq(coeff(P[n], t, i), i = 0 .. degree(P[n])) end do; # yields sequence in triangular form - Emeric Deutsch, Nov 12 2015

Mathematica

T[n_, k_] := SeriesCoefficient[QPochhammer[1-t, x^2]/(t*QPochhammer[x]), {x, 0, n}, {t, 0, k}]; Table[DeleteCases[Table[T[n, k], {k, 0, n}], 0], {n, 0, 25}] // Flatten (* Jean-François Alcover, Dec 11 2016 *)

Crossrefs

Cf. A000009, A000041, A024786, A090867, A259361.

Sequence in context: A102885 A323089 A239511 * A138585 A070048 A116498

Adjacent sequences: A264049 A264050 A264051 * A264053 A264054 A264055

Keyword

nonn,look,tabf

Author

Christian Stump, Nov 01 2015

Extensions

More terms from Alois P. Heinz, Nov 02 2015

Status

approved