%I A060524
%S A060524 1,0,1,1,0,1,0,5,0,1,9,0,14,0,1,0,89,0,30,0,1,225,0,439,0,55,0,1,0,
%T A060524 3429,0,1519,0,91,0,1,11025,0,24940,0,4214,0,140,0,1,0,230481,0,122156,
%U A060524 0,10038,0,204,0,1,893025,0,2250621,0,463490,0,21378,0,285,0,1,0
%N A060524 Triangle T(n,k) giving number of degree-n permutations with k odd cycles, 
               k=0..n.
%C A060524 The row polynomials t(n,x):=sum(T(n,k)*x^k,k=0..n) satisfy the recurrence 
               relation t(n,x)= x*t(n-1,x) + ((n-1)^2)*t(n-2,x); t(-1,x)=0,t(0,x)=1. 
               W. Lang, see above.
%C A060524 This is an example of a Sheffer triangle (coefficient triangle for Sheffer 
               polynomials). In the umbral calculus (see the Roman reference given 
               under A048854) s(n,x) := sum(T(n,k)*x^k,k=0..n) would be called Sheffer 
               polynomials for (1/cosh(t),tanh(t)), which translates to the e.g.f. 
               for column nr. k>=0 given by (1/sqrt(1-x^2))*((artanh(x))^k)/k!. 
               The e.g.f. given below is rewritten in this Sheffer context as (1/
               sqrt(1-x^2))*exp(y*ln(sqrt((1+x)/(1-x))))= (1/sqrt(1-x^2))*exp(y*artanh(x)). 
               The rows of the Jabotinsky type triangle |A049218| provide the coefficients 
               of the associated polynomials. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Feb 24 2005.
%C A060524 The solution of the differential-difference relation f(n+1,x)= diff(f(n,
               x),x) + (n^2)*f(n-1,x), n>=1, with inputs f(0,x) and f(1,x)=diff(f(0,
               x),x) is f(n,x)= t(n,d_x)*f(0,x), with the differential operator 
               d_x:=d/dx and the row polynomials t(n,x) defined above. This problem 
               appears in a computation of thermo field dynamics where f(0,x)=1/
               cosh(x). See the triangle A060081. W. Lang, see above.
%C A060524 The inverse of the Sheffer matrix T with elements T(n,k) is the Sheffer 
               matrix A060081. - W. Lang, Jul 22 2005
%C A060524 T(n,k)=0 if n-k= 1(mod 2), else T(n,k)= sum of M2(n,p), p from {1,...,
               A000041(n)} restricted to partitions with exactly k odd parts and 
               any nonnegative number of even parts. For the M2-multinomial numbers 
               in A-St order see A036039(n,p). W. Lang, Aug 07 2007.
%F A060524 E.g.f.: (1+x)^((y-1)/2)/(1-x)^((y+1)/2).
%F A060524 T(n, k) = T(n-1, k-1) + ((n-1)^2)*T(n-2, k); T(-1, k):=0, T(n, -1):=0, 
               T(0, 0)=1, T(n, k)=0 if n<k. W. Lang, see above.
%e A060524 [1], [0, 1], [1, 0, 1], [0, 5, 0, 1], [9, 0, 14, 0, 1], [0, 89, 0, 30, 
               0, 1], [225, 0, 439, 0, 55, 0, 1], [0, 3429, 0, 1519, 0, 91, 0, 1], 
               [11025, 0, 24940, 0, 4214, 0, 140, 0, 1], [0, 230481, 0, 122156, 
               0, 10038, 0, 204, 0, 1], [893025, 0, 2250621, 0, 463490, 0, 21378, 
               0, 285, 0, 1], [0, 23941125, 0, 14466221, 0, 1467290, 0, 41778, 0, 
               385, 0, 1], ...
%Y A060524 Cf. A060338, A060523.
%Y A060524 A111594 (associated Sheffer polynomials).
%Y A060524 Sequence in context: A097591 A164652 A127557 this_sequence A133843 A064315 
               A099221
%Y A060524 Adjacent sequences: A060521 A060522 A060523 this_sequence A060525 A060526 
               A060527
%K A060524 easy,nonn,tabl
%O A060524 0,8
%A A060524 Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 01 2001