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A275281 Number T(n,k) of set partitions of [n] with symmetric block size list of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 13
1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 7, 0, 1, 0, 1, 10, 19, 13, 3, 1, 0, 1, 0, 56, 0, 22, 0, 1, 0, 1, 35, 160, 171, 86, 34, 4, 1, 0, 1, 0, 463, 0, 470, 0, 50, 0, 1, 0, 1, 126, 1337, 2306, 2066, 1035, 250, 70, 5, 1, 0, 1, 0, 3874, 0, 10299, 0, 2160, 0, 95, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

Wikipedia, Partition of a set

FORMULA

T(n,k) = 0 if n is odd and k is even.

EXAMPLE

T(4,2) = 3: 12|34, 13|24, 14|23.

T(5,3) = 7: 12|3|45, 13|2|45, 1|234|5, 1|235|4, 14|2|35, 1|245|3, 15|2|34.

T(6,4) = 13: 12|3|4|56, 13|2|4|56, 1|23|45|6, 1|23|46|5, 14|2|3|56, 1|24|35|6, 1|24|36|5, 1|25|34|6, 1|26|34|5, 15|2|3|46, 1|25|36|4, 1|26|35|4, 16|2|3|45.

T(7,5) = 22: 12|3|4|5|67, 13|2|4|5|67, 1|23|4|56|7, 1|23|4|57|6, 14|2|3|5|67, 1|24|3|56|7, 1|24|3|57|6, 1|2|345|6|7, 1|2|346|5|7, 1|2|347|5|6, 15|2|3|4|67, 1|25|3|46|7, 1|25|3|47|6, 1|2|356|4|7, 1|2|357|4|6, 1|26|3|45|7, 1|27|3|45|6, 16|2|3|4|57, 1|26|3|47|5, 1|2|367|4|5, 1|27|3|46|5, 17|2|3|4|56.

Triangle T(n,k) begins:

  1;

  0, 1;

  0, 1,   1;

  0, 1,   0,    1;

  0, 1,   3,    2,    1;

  0, 1,   0,    7,    0,    1;

  0, 1,  10,   19,   13,    3,    1;

  0, 1,   0,   56,    0,   22,    0,   1;

  0, 1,  35,  160,  171,   86,   34,   4,  1;

  0, 1,   0,  463,    0,  470,    0,  50,  0, 1;

  0, 1, 126, 1337, 2306, 2066, 1035, 250, 70, 5, 1;

MAPLE

b:= proc(n, s) option remember; expand(`if`(n>s,

      binomial(n-1, n-s-1)*x, 1)+add(binomial(n-1, j-1)*

      b(n-j, s+j)*binomial(s+j-1, j-1), j=1..(n-s)/2)*x^2)

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):

seq(T(n), n=0..12);

MATHEMATICA

b[n_, s_] := b[n, s] = Expand[If[n>s, Binomial[n-1, n-s-1]*x, 1] + Sum[ Binomial[n-1, j-1]*b[n-j, s+j]*Binomial[s+j-1, j-1], {j, 1, (n-s)/2} ]*x^2]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, Feb 03 2017, translated from Maple *)

CROSSREFS

Columns k=0-1,3,5,7,9 give: A000007, A000012 for n>0, A275289, A275290, A275291, A275292.

Bisections of columns k=2,4,6,8,10 give: A001700(n-1) for n>0, A275293, A275294, A275295, A275296.

Row sums give A275282.

T(n,A004525(n)) gives A305197.

T(2n,n) gives A275283.

T(2n+1,A109613(n)) gives A305198.

T(n,n) gives A000012.

T(n+3,n+1) gives A002623.

Sequence in context: A071960 A056898 A204065 * A204176 A062160 A301296

Adjacent sequences:  A275278 A275279 A275280 * A275282 A275283 A275284

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 21 2016

STATUS

approved

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Last modified March 18 17:51 EDT 2019. Contains 321292 sequences. (Running on oeis4.)