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

0,4

Comments

Triangle is related to the tangent numbers A000182.

References

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

Links

Table of n, a(n) for n=0..56.

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)|.

Example

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;
  ...

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

Crossrefs

Similar to A104035. Leading edge is essentially A000182.

Cf. A003707.

Sequence in context: A068463 A099554 A319697 * A363025 A300073 A113400

Adjacent sequences: A107726 A107727 A107728 * A107730 A107731 A107732

Keyword

nonn,easy,tabl

Author

N. J. A. Sloane, Jun 10 2005

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

Status

approved