%I A099174
%S A099174 1,0,1,1,0,1,0,3,0,1,3,0,6,0,1,0,15,0,10,0,1,15,0,45,0,15,0,1,0,105,0,
%T A099174 105,0,21,0,1,105,0,420,0,210,0,28,0,1,0,945,0,1260,0,378,0,36,0,1,945,
%U A099174 0,4725,0,3150,0,630,0,45,0,1,0,10395,0,17325,0,6930,0,990,0,55
%N A099174 Triangle read by rows: coefficients of modified Hermite polynomials.
%C A099174 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
%C A099174 Riordan array [exp(x^2/2),x]. [From Paul Barry (pbarry(AT)wit.ie), Nov 
               06 2008]
%H A099174 A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, 
               <a href="http://arXiv.org/abs/quant-ph/0409152">A product formula 
               and combinatorial field theory</a>
%F A099174 h(k, x) = (-I/sqrt(2))^k * H(k, I*x/sqrt(2)), H(n, x) the Hermite polynomials 
               (A060821, A059343).
%F A099174 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
%F A099174 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]
%e A099174 h(0,x) = 1
%e A099174 h(1,x) = x
%e A099174 h(2,x) = x^2 + 1
%e A099174 h(3,x) = x^3 + 3*x
%e A099174 h(4,x) = x^4 + 6*x^2 + 3
%e A099174 h(5,x) = x^5 + 10*x^3 + 15*x
%e A099174 h(6,x) = x^6 + 15*x^4 + 45*x^2 + 15
%e A099174 Contribution from Paul Barry (pbarry(AT)wit.ie), Nov 06 2008: (Start)
%e A099174 Triangle begins
%e A099174 1,
%e A099174 0, 1,
%e A099174 1, 0, 1,
%e A099174 0, 3, 0, 1,
%e A099174 3, 0, 6, 0, 1,
%e A099174 0, 15, 0, 10, 0, 1,
%e A099174 15, 0, 45, 0, 15, 0, 1
%e A099174 Production array starts
%e A099174 0, 1,
%e A099174 1, 0, 1,
%e A099174 0, 2, 0, 1,
%e A099174 0, 0, 3, 0, 1,
%e A099174 0, 0, 0, 4, 0, 1,
%e A099174 0, 0, 0, 0, 5, 0, 1 (End)
%p A099174 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
%Y A099174 Row sums (values at 1) are A000085. Values at 2 are A005425.
%Y A099174 Sequence in context: A035653 A126595 A066325 this_sequence A137297 A095710 
               A160052
%Y A099174 Adjacent sequences: A099171 A099172 A099173 this_sequence A099175 A099176 
               A099177
%K A099174 nonn,tabl
%O A099174 0,8
%A A099174 Ralf Stephan, on a suggestion of Karol A. Penson, Oct 13 2004