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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).

%I A260338 #20 Feb 17 2016 08:20:12
%S A260338 1,1,1,2,2,2,5,6,6,6,17,23,24,24,24,73,109,118,120,120,120,388,618,
%T A260338 690,714,720,720,720,2461,4096,4686,4926,5016,5040,5040,5040,18155,
%U A260338 31133,36308,38688,39768,40200,40320,40320,40320
%N 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).
%H A260338 Alois P. Heinz, <a href="/A260338/b260338.txt">Rows n = 0..140, flattened</a>
%H A260338 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 A260338 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 A260338 Triangle begins:
%e A260338 1,
%e A260338 1,1,
%e A260338 2,2,2,
%e A260338 5,6,6,6,
%e A260338 17,23,24,24,24,
%e A260338 73,109,118,120,120,120,
%e A260338 388,618,690,714,720,720,720,
%e A260338 ...
%p A260338 phi:= proc(m, n) option remember;
%p A260338         `if`(n+m<2, 1, `if`(n+m=2, 2, m*phi(m-1, n)-
%p A260338         `if`(n=0, (m-1)*(m-2)/2*phi(m-3, 0), -n*phi(m, n-1))))
%p A260338       end:
%p A260338 seq(seq(phi(m-k, k), k=0..m), m=0..10);  # _Alois P. Heinz_, Jul 30 2015
%t A260338 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 A260338 The outer diagonals are A002135, A000142.
%K A260338 nonn,tabl
%O A260338 0,4
%A A260338 _N. J. A. Sloane_, Jul 30 2015
%E A260338 More terms from _Alois P. Heinz_, Jul 30 2015