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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.

%I A275281 #23 Nov 27 2024 11:56:46
%S A275281 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,
%T A275281 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,
%U A275281 1337,2306,2066,1035,250,70,5,1,0,1,0,3874,0,10299,0,2160,0,95,0,1
%N 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.
%H A275281 Alois P. Heinz, <a href="/A275281/b275281.txt">Rows n = 0..200, flattened</a>
%H A275281 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A275281 T(n,k) = 0 if n is odd and k is even.
%e A275281 T(4,2) = 3: 12|34, 13|24, 14|23.
%e A275281 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.
%e A275281 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.
%e A275281 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.
%e A275281 Triangle T(n,k) begins:
%e A275281   1;
%e A275281   0, 1;
%e A275281   0, 1,   1;
%e A275281   0, 1,   0,    1;
%e A275281   0, 1,   3,    2,    1;
%e A275281   0, 1,   0,    7,    0,    1;
%e A275281   0, 1,  10,   19,   13,    3,    1;
%e A275281   0, 1,   0,   56,    0,   22,    0,   1;
%e A275281   0, 1,  35,  160,  171,   86,   34,   4,  1;
%e A275281   0, 1,   0,  463,    0,  470,    0,  50,  0, 1;
%e A275281   0, 1, 126, 1337, 2306, 2066, 1035, 250, 70, 5, 1;
%e A275281   ...
%p A275281 b:= proc(n, s) option remember; expand(`if`(n>s,
%p A275281       binomial(n-1, n-s-1)*x, 1)+add(binomial(n-1, j-1)*
%p A275281       b(n-j, s+j)*binomial(s+j-1, j-1), j=1..(n-s)/2)*x^2)
%p A275281     end:
%p A275281 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
%p A275281 seq(T(n), n=0..12);
%t A275281 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 *)
%Y A275281 Columns k=0-1,3,5,7,9 give: A000007, A000012 for n>0, A275289, A275290, A275291, A275292.
%Y A275281 Bisections of columns k=2,4,6,8,10 give: A001700(n-1) for n>0, A275293, A275294, A275295, A275296.
%Y A275281 Row sums give A275282.
%Y A275281 T(n,A004525(n)) gives A305197.
%Y A275281 T(2n,n) gives A275283.
%Y A275281 T(2n+1,A109613(n)) gives A305198.
%Y A275281 T(n,n) gives A000012.
%Y A275281 T(n+3,n+1) gives A002623.
%K A275281 nonn,tabl
%O A275281 0,13
%A A275281 _Alois P. Heinz_, Jul 21 2016