login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 16 17:49 EDT 2019. Contains 328102 sequences. (Running on oeis4.)