A259465 Triangle read by rows: enumerates pairs of amicable permutations by rises.

1, 1, 1, 1, 1, 8, 1, 1, 43, 43, 1, 1, 194, 826, 194, 1, 1, 803, 11284, 11284, 803, 1, 1, 3184, 127905, 392244, 127905, 3184, 1, 1, 12367, 1297629, 10258067, 10258067, 1297629, 12367, 1, 1, 47606, 12295720, 224702858, 561134638, 224702858, 12295720, 47606, 1

Offset: 0

Author

N. J. A. Sloane, Jun 30 2015, following a suggestion from L. Carlitz, Nov 30 1975

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  1;
  1;
  1,     1;
  1,     8,       1;
  1,    43,      43,        1;
  1,   194,     826,      194,        1;
  1,   803,   11284,    11284,      803,       1;
  1,  3184,  127905,   392244,   127905,    3184,     1;
  1, 12367, 1297629, 10258067, 10258067, 1297629, 12367, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..80, flattened
• Carlitz, L., Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bulletin of the American Mathematical Society 80.5 (1974): 881-884. [Annotated scanned copy]
• L. Carlitz, N. J. A. Sloane, and C. L. Mallows, Correspondence, 1975

Crossrefs

Row sums give A060350.

Programs

Maple

b:= proc(u, o, h) option remember; `if`(u+o=0, 1, expand(
      add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h)*x, j=1..u)+
      add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(0, n$2)):
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 02 2015

Mathematica

b[u_, o_, h_] := b[u, o, h] = If[u+o == 0, 1, Expand[Sum[Sum[b[u-j, o+j-1, h+i-1], {i, 1, u+o-h}]*x, {j, 1, u}] + Sum[Sum[b[u+j-1, o-j, h-i], {i, 1, h}], {j, 1, o}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[0, n, n]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 12 2016, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Jul 02 2015