A260338 Triangle read by rows: Cayley's numbers phi(m,n) (m,n>=0). Row m contains phi(m,0), phi(m-1,1), phi(m-2,2), ..., phi(0,m).

1, 1, 1, 2, 2, 2, 5, 6, 6, 6, 17, 23, 24, 24, 24, 73, 109, 118, 120, 120, 120, 388, 618, 690, 714, 720, 720, 720, 2461, 4096, 4686, 4926, 5016, 5040, 5040, 5040, 18155, 31133, 36308, 38688, 39768, 40200, 40320, 40320, 40320

Offset

0,4

Links

Alois P. Heinz, Rows n = 0..140, flattened

A. Cayley, On the number of distinct terms in a symmetrical or partially symmetrical determinant, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, pp. 185-190. [Annotated scanned copy]

Formula

phi(0,0)=1, phi(1,0)=phi(0,1)=1, phi(2,0)=phi(1,1)=phi(0,2)=2; for m>2, phi(m,0) = m*phi(m-1,0) - (m-1)*(m-2)/2*phi(m-3,0); for m>2, n>0, phi(m,n) = m*phi(m-1,n) + n*phi(m,n-1).

Example

Triangle begins:
1,
1,1,
2,2,2,
5,6,6,6,
17,23,24,24,24,
73,109,118,120,120,120,
388,618,690,714,720,720,720,
...

Maple

phi:= proc(m, n) option remember;
        `if`(n+m<2, 1, `if`(n+m=2, 2, m*phi(m-1, n)-
        `if`(n=0, (m-1)*(m-2)/2*phi(m-3, 0), -n*phi(m, n-1))))
      end:
seq(seq(phi(m-k, k), k=0..m), m=0..10);  # Alois P. Heinz, Jul 30 2015

Mathematica

phi[m_, n_] := phi[m, n] = If[n+m < 2, 1, If[n+m == 2, 2, m*phi[m-1, n] - If[n == 0, (m-1)*(m-2)/2*phi[m-3, 0], -n*phi[m, n-1]]]]; Table[Table[phi[ m-k, k], {k, 0, m}], {m, 0, 10}] // Flatten (* Jean-François Alcover, Feb 17 2016, after Alois P. Heinz *)

Crossrefs

The outer diagonals are A002135, A000142.

Sequence in context: A033306 A136347 A279515 * A023569 A263373 A145876

Adjacent sequences: A260335 A260336 A260337 * A260339 A260340 A260341

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jul 30 2015

Extensions

More terms from Alois P. Heinz, Jul 30 2015

Status

approved