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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
OFFSET
0,13
LINKS
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-François 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
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 21 2016
STATUS
approved