A082635 Square array read by antidiagonals: degree of the K(2,p)^q variety.

1, 2, 1, 5, 8, 1, 14, 55, 32, 1, 42, 364, 610, 128, 1, 132, 2380, 9842, 6765, 512, 1, 429, 15504, 147798, 265720, 75025, 2048, 1, 1430, 100947, 2145600, 9112264, 7174454, 832040, 8192, 1, 4862, 657800, 30664890, 290926848, 562110290, 193710244

Offset

1,2

Comments

Numbers are related to the dynamic pole assignment problem. "The variety K(m,p)^q can also be viewed as the parameterization of the space of rational curves of degree q of the Grassmann variety Grass(m,m+p)".

Also lim(n->inf, T(n+1,2i)/T(n,2i)) = 4^(i+1).

Links

Table of n, a(n) for n=1..42.

M. S. Ravi et al., Dynamic pole assignment and Schubert calculus, SIAM J. Control Optimization, 34 (1996), 813-832, esp. p. 825.

Formula

degK2(p, q) = (-1)^q * (2*p+p*q+2*q)! * Sum_{j=0..q} ((q-2*j) * (p+2)+1) / (p+j*(p+2))! / (p+1+(q-j)*(p+2))!.

Example

Top left corner of array:
  1,   2,     5,      14,      42,    132,     429, 1430, ... A000108 (Catalan numbers);
  1,   8,    55,     364,    2380,   15504, 100947,  ...      A013068 deg K(2,n)^1;
  1,  32,   610,    9842,  147798, 2145600,    ...,           A013069 deg K(2,n)^2;
  1, 128,  6765,  265720, 9112264,     ...,                   A013070 deg K(2,n)^3;
  1, 512, 75025, 7174454,     ...,                            A013071 deg K(2,n)^4;
  ...

Crossrefs

Cf. A013702.

Second column is A004171(q), third is A000045(5q).

T(n, 2*i) = A080934((i+1)*n+2*i, n+1).

Sequence in context: A141507 A193603 A059274 * A377661 A166623 A348729

Adjacent sequences: A082632 A082633 A082634 * A082636 A082637 A082638

Keyword

nonn,tabl,easy

Author

Ralf Stephan, May 14 2003

Status

approved