0,8

T(n,k) is the number of involutions of {1,2,...,n}, having k fixed points (0<=k<=n). Example: T(4,2)=6 because we have 1243,1432,1324,4231,3214 and 2134. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 14 2006

Riordan array [exp(x^2/2),x]. [From Paul Barry (pbarry(AT)wit.ie), Nov 06 2008]

A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory

h(k, x) = (-I/sqrt(2))^k * H(k, I*x/sqrt(2)), H(n, x) the Hermite polynomials (A060821, A059343).

T(n,k)=n!/[2^((n-k)/2)*((n-k)/2)!k! ] if n-k>=0 is even; 0 otherwise. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 14 2006

G.f.: 1/(1-x*y-x^2/(1-x*y-2*x^2/(1-x*y-3*x^2/(1-x*y-4*x^2/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Apr 10 2009]

h(0,x) = 1

h(1,x) = x

h(2,x) = x^2 + 1

h(3,x) = x^3 + 3*x

h(4,x) = x^4 + 6*x^2 + 3

h(5,x) = x^5 + 10*x^3 + 15*x

h(6,x) = x^6 + 15*x^4 + 45*x^2 + 15

Contribution from Paul Barry (pbarry(AT)wit.ie), Nov 06 2008: (Start)

Triangle begins

1,

0, 1,

1, 0, 1,

0, 3, 0, 1,

3, 0, 6, 0, 1,

0, 15, 0, 10, 0, 1,

15, 0, 45, 0, 15, 0, 1

Production array starts

0, 1,

1, 0, 1,

0, 2, 0, 1,

0, 0, 3, 0, 1,

0, 0, 0, 4, 0, 1,

0, 0, 0, 0, 5, 0, 1 (End)

T:=proc(n, k) if n-k mod 2 = 0 then n!/2^((n-k)/2)/((n-k)/2)!/k! else 0 fi end: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 14 2006

Row sums (values at 1) are A000085. Values at 2 are A005425.

Adjacent sequences: A099171 A099172 A099173 this_sequence A099175 A099176 A099177

Sequence in context: A035653 A126595 A066325 this_sequence A137297 A095710 A160052

nonn,tabl

Ralf Stephan, on a suggestion of Karol A. Penson, Oct 13 2004