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A060524 Triangle read by rows: T(n,k) = number of degree-n permutations with k odd cycles, k=0..n, n >= 0. 13
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
The row polynomials t(n,x):=Sum_{k=0..n} T(n,k)*x^k 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. - Wolfdieter Lang, see above.
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_{k=0..n} T(n,k)*x^k would be called Sheffer polynomials for (1/cosh(t),tanh(t)), which translates to the e.g.f. for column number k>=0 given by (1/sqrt(1-x^2))*((arctanh(x))^k)/k!. The e.g.f. given below is rewritten in this Sheffer context as (1/sqrt(1-x^2))*exp(y*log(sqrt((1+x)/(1-x))))= (1/sqrt(1-x^2))*exp(y*arctanh(x)). The rows of the Jabotinsky type triangle |A049218| provide the coefficients of the associated polynomials. - Wolfdieter Lang, Feb 24 2005
The solution of the differential-difference relation f(n+1,x)= (d/dx)f(n,x) + (n^2)*f(n-1,x), n >= 1, with inputs f(0,x) and f(1,x) = (d/dx)f(0,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. - Wolfdieter Lang, Feb 24 2005
The inverse of the Sheffer matrix T with elements T(n,k) is the Sheffer matrix A060081. - Wolfdieter Lang, Jul 22 2005
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). - Wolfdieter Lang, Aug 07 2007
LINKS
A. Hamdi and J. Zeng. Orthogonal polynomials and operator orderings. J. Math. Phys., 51:043506, (2010); arXiv:1006.0808 [math.CO], 2010.
FORMULA
E.g.f.: (1+x)^((y-1)/2)/(1-x)^((y+1)/2).
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. - Wolfdieter Lang, see above.
The Meixner polynomials defined by S_0(x)=1, S_1(x) = x; S_{n+1}(x) = x*S_n(x) - n^2*S_{n-1}(x) give a signed version of this triangle (cf. A060338). - N. J. A. Sloane, May 30 2013
EXAMPLE
Triangle begins:
[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],
...
Signed version begins:
[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],
...
MAPLE
with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
add(multinomial(n, n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1)*
`if`(irem(i, 2)=1, x^j, 1), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12); # Alois P. Heinz, Mar 09 2015
MATHEMATICA
nn = 6; Range[0, nn]! CoefficientList[
Series[(1 - x^2)^(-1/2) ((1 + x)/(1 - x))^(y/2), {x, 0, nn}], {x, y}] // Grid (* Geoffrey Critzer, Aug 28 2012 *)
CROSSREFS
Cf. A111594 (associated Sheffer polynomials).
Sequence in context: A318299 A164652 A127557 * A133843 A215083 A221308
KEYWORD
easy,nonn,tabl
AUTHOR
Vladeta Jovovic, Apr 01 2001
STATUS
approved

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Last modified June 5 11:36 EDT 2023. Contains 363136 sequences. (Running on oeis4.)