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A082635 Square array read by antidiagonals: degree of the K(2,p)^q variety. 0


%S 1,2,1,5,8,1,14,55,32,1,42,364,610,128,1,132,2380,9842,6765,512,1,429,

%T 15504,147798,265720,75025,2048,1,1430,100947,2145600,9112264,7174454,

%U 832040,8192,1,4862,657800,30664890,290926848,562110290,193710244

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

%C 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)".

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

%H M. S. Ravi et al., <a href="http://www.nd.edu/~rosen/Paper/degree.ps">Dynamic pole assignment and Schubert calculus</a>, SIAM J. Control Optimization, 34 (1996), 813-832, esp. p. 825.

%F degK2(p, q)=(-1)^q*(2p+pq+2q)!*sum(j=0, q, ((q-2j)(p+2)+1)/(p+j(p+2))!/(p+1+(q-j)(p+2))!).

%e Top left corner of array:

%e 1,2,5,14,42,132,429,1430,... A000108 (Catalan numbers)

%e 1,8,55,364,2380,15504,100947,...A013068 deg K(2,n)^1

%e 1,32,610,9842,147798,2145600,...A013069 deg K(2,n)^2

%e 1,128,6765,265720,9112264,... A013070 deg K(2,n)^3

%e 1,512,75025,7174454,... A013071 deg K(2,n)^4

%Y Cf. A013702.

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

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

%K nonn,tabl,easy

%O 1,2

%A _Ralf Stephan_, May 14 2003

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