A395326 Triangle read by rows: T(n,k) is the number of partitions of {1..2n} that are invariant under a permutation consisting of n 2-cycles with k orbits of blocks.

1, 0, 2, 0, 3, 4, 0, 5, 18, 8, 0, 9, 67, 72, 16, 0, 17, 240, 470, 240, 32, 0, 33, 859, 2745, 2420, 720, 64, 0, 65, 3108, 15323, 20790, 10360, 2016, 128, 0, 129, 11407, 84042, 165081, 122640, 39200, 5376, 256, 0, 257, 42480, 458900, 1257480, 1307922, 610848, 135744, 13824, 512

Offset

0,3

Links

Alois P. Heinz, Rows n = 0..150, flattened

Formula

E.g.f.: exp( y*((exp(2*x) - 1)/2 + exp(x) - 1 ) ).

Example

Triangle begins:
  1;
  0,   2;
  0,   3,     4;
  0,   5,    18,     8;
  0,   9,    67,    72,     16;
  0,  17,   240,   470,    240,     32;
  0,  33,   859,  2745,   2420,    720,    64;
  0,  65,  3108, 15323,  20790,  10360,  2016,  128;
  0, 129, 11407, 84042, 165081, 122640, 39200, 5376, 256;
  ...

Maple

b:= proc(n) option remember; `if`(n=0, 1, add((1+
      2^(j-1))*binomial(n-1, j-1)*expand(b(n-j)*x), j=1..n))
    end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n, k), k=0..n), n=0..9);  # Alois P. Heinz, Apr 27 2026

Prog

(PARI) T(n) = {[Vecrev(p) | p<-Vec(serlaplace(exp( y*((exp(2*x + O(x*x^n)) - 1)/2 + exp(x + O(x*x^n)) - 1 ) )))]}
{ my(A=T(8)); for(i=1, #A, print(A[i])) }

Crossrefs

Row sums are A002872.

Main diagonal is A000079.

Column k=1 is A000051(n-1).

Cf. A293181, A395327.

Sequence in context: A261094 A091538 A340991 * A013584 A307320 A350227

Adjacent sequences: A395323 A395324 A395325 * A395327 A395328 A395329

Keyword

nonn,tabl

Author

Andrew Howroyd and Andrei Zabolotskii, Apr 19 2026

Status

approved