A107729 Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(0,0) = 1; T(0,k) = 0 if k < 0 or if k > 0; T(n,k) = k*T(n-1,k-1) + (k+2)*T(n-1,k+1).

1, 0, 1, 2, 0, 2, 0, 8, 0, 6, 16, 0, 40, 0, 24, 0, 136, 0, 240, 0, 120, 272, 0, 1232, 0, 1680, 0, 720, 0, 3968, 0, 12096, 0, 13440, 0, 5040, 7936, 0, 56320, 0, 129024, 0, 120960, 0, 40320, 0, 176896, 0, 814080, 0, 1491840, 0, 1209600, 0, 362880, 353792, 0

Offset: 0

Author

N. J. A. Sloane, Jun 10 2005

Keywords

• nonn
• easy
• tabl

Comments

Triangle is related to the tangent numbers A000182.

Examples

			Triangle begins:
     1;
     0,    1;
     2,    0,    2;
     0,    8,    0,    6;
    16,    0,   40,    0,    24;
     0,   136,   0,   240,    0,   120;
    272,   0,  1232,   0,   1680,   0,    720;
     0,  3968,   0,  12096,   0,  13440,   0,  5040;
   7936,   0,  56320,  0,  129024,  0,  120960,  0,   40320;
     0, 176896,  0, 814080,   0, 1491840,  0, 1209600,  0, 362880;
  353792,  0, 3610112, 0, 12207360, 0, 18627840, 0, 13305660, 0, 3628800;
  ...
		

References

• S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 446.

Crossrefs

Similar to A104035. Leading edge is essentially A000182.

Cf. A003707.

Programs

Maple

T:=proc(n,k) if k=-1 then 0 elif n=1 and k=1 then 1 elif k>n then 0 else (k-1)*T(n-1,k-1)+(k+1)*T(n-1,k+1) fi end: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form [Produces triangle with a different offset] # Emeric Deutsch, Jun 13 2005

Formula

T(n, n) = n!; T(n, 0) = 0 if n = 2m+1; T(n, 0) = A000182(m+1) if n = 2m.

Sum_{k>=0} T(m, k)*T(n, k)*(k+1) = T(m+n, 0).

Sum_{k>=0} T(n, k) = |A003707(n+1)|.

Extensions

More terms from Emeric Deutsch, Jun 13 2005

Additional comments from Philippe Deléham, Sep 17 2005

Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar