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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).
2

%I #20 Feb 17 2016 08:20:12

%S 1,1,1,2,2,2,5,6,6,6,17,23,24,24,24,73,109,118,120,120,120,388,618,

%T 690,714,720,720,720,2461,4096,4686,4926,5016,5040,5040,5040,18155,

%U 31133,36308,38688,39768,40200,40320,40320,40320

%N 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).

%H Alois P. Heinz, <a href="/A260338/b260338.txt">Rows n = 0..140, flattened</a>

%H A. Cayley, <a href="/A260338/a260338.pdf">On the number of distinct terms in a symmetrical or partially symmetrical determinant</a>, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, pp. 185-190. [Annotated scanned copy]

%F 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).

%e Triangle begins:

%e 1,

%e 1,1,

%e 2,2,2,

%e 5,6,6,6,

%e 17,23,24,24,24,

%e 73,109,118,120,120,120,

%e 388,618,690,714,720,720,720,

%e ...

%p phi:= proc(m, n) option remember;

%p `if`(n+m<2, 1, `if`(n+m=2, 2, m*phi(m-1, n)-

%p `if`(n=0, (m-1)*(m-2)/2*phi(m-3, 0), -n*phi(m, n-1))))

%p end:

%p seq(seq(phi(m-k, k), k=0..m), m=0..10); # _Alois P. Heinz_, Jul 30 2015

%t 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_ *)

%Y The outer diagonals are A002135, A000142.

%K nonn,tabl

%O 0,4

%A _N. J. A. Sloane_, Jul 30 2015

%E More terms from _Alois P. Heinz_, Jul 30 2015