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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.

%I A060524 #69 Jul 30 2024 09:26:23
%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,
%U A060524 122156,0,10038,0,204,0,1,893025,0,2250621,0,463490,0,21378,0,285,0,1,0
%N A060524 Triangle read by rows: T(n,k) = number of degree-n permutations with k odd cycles, k=0..n, n >= 0.
%C A060524 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.
%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_{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
%C A060524 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
%C A060524 The inverse of the Sheffer matrix T with elements T(n,k) is the Sheffer matrix A060081. - _Wolfdieter 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). - _Wolfdieter Lang_, Aug 07 2007
%H A060524 Alois P. Heinz, <a href="/A060524/b060524.txt">Rows n = 0..140, flattened</a>
%H A060524 Marcelo Aguiar, Sarah Brauner, and Vic Reiner, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2024/13.pdf">Configuration spaces and peak representations</a>, Sém. Lotharingien Comb., Proc. 36th Conf. Formal Power Series Alg. Comb. (2024) Vol. 91B, Art. No. 13. See p. 12.
%H A060524 Peter Bala, <a href="/A060524/a060524_3.pdf">Meixner polynomials and Brouncker's continued fraction for Pi</a>
%H A060524 Paul L. Butzer and Tom H. Koornwinder, <a href="https://doi.org/10.1016/j.indag.2018.09.009">Josef Meixner: His life and his orthogonal polynomials</a>, Indagationes Mathematicae, Volume 30, Issue 1, January 2019, Pages 250-264.
%H A060524 Adel Hamdi and Jiang Zeng, <a href="http://arxiv.org/abs/1006.0808">Orthogonal polynomials and operator orderings</a>, J. Math. Phys., 51:043506, (2010); arXiv:1006.0808 [math.CO], 2010.
%H A060524 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MeixnerPolynomialoftheFirstKind.html">Meixner polynomial of the first kind</a>.
%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. - _Wolfdieter Lang_, see above.
%F A060524 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
%F A060524 From _Peter Bala_, Apr 10 2024: (Start)
%F A060524 The n-th row polynomial R(n, x) satisfies
%F A060524 (4*n + 2)*R(n, x) = (x + 1)*R(n, x+2) - (x - 1)*R(n, x-2).
%F A060524 Series for Pi involving the row polynomials R(n, x): for n >= 0 there holds
%F A060524 ((2*n + 1)!!/(2^n*n!))^2 * Pi = (4*n + 3) + 4*((2*n + 1)!!^4) * Sum_{k >= 1} (-1)^(k+1)/((2*k + 1)*R(2*n+1, 2*k)*R(2*n+1, 2*k+2)). Cf. A142979 and A142983.
%F A060524 R(2*n, 0) = A001147(n)^2 = A001818(n); R(2*n+1, 0) = 0.
%F A060524 R(n, 1) = n! = A000142(n).
%F A060524 R(2*n, 2) = (4*n + 1)*A001147(n)^2 = (4*n + 1)*((2*n)!/(2^n*n!))^2;
%F A060524 R(2*n+1, 2) = 2*A001447(n+1)^2 = 2*(2*n + 1)!^2/(n!^2*4^n).
%F A060524 R(n, 3) = (2*n + 1)*n! = A007680(n). (End)
%e A060524 Triangle begins:
%e A060524   [1],
%e A060524   [0, 1],
%e A060524   [1, 0, 1],
%e A060524   [0, 5, 0, 1],
%e A060524   [9, 0, 14, 0, 1],
%e A060524   [0, 89, 0, 30, 0, 1],
%e A060524   [225, 0, 439, 0, 55, 0, 1],
%e A060524   [0, 3429, 0, 1519, 0, 91, 0, 1],
%e A060524   [11025, 0, 24940, 0, 4214, 0, 140, 0, 1],
%e A060524   [0, 230481, 0, 122156, 0, 10038, 0, 204, 0, 1],
%e A060524   [893025, 0, 2250621, 0, 463490, 0, 21378, 0, 285, 0, 1],
%e A060524   [0, 23941125, 0, 14466221, 0, 1467290, 0, 41778, 0, 385, 0, 1],
%e A060524   ...
%e A060524 Signed version begins:
%e A060524   [1],
%e A060524   [0, 1],
%e A060524   [-1, 0, 1],
%e A060524   [0, -5, 0, 1],
%e A060524   [9, 0, -14, 0, 1],
%e A060524   [0, 89, 0, -30, 0, 1],
%e A060524   [-225, 0, 439, 0, -55, 0, 1],
%e A060524   [0, -3429, 0, 1519, 0, -91, 0, 1],
%e A060524   ...
%e A060524 From _Peter Bala_, Feb 23 2024: (Start)
%e A060524 Maple can verify the following series for Pi:
%e A060524 Row 1 polynomial R(1, x) = x:
%e A060524 Pi = 3 + 4*Sum_{n >= 1} (-1)^(n+1)/((2*n + 1)*R(1, 2*n)*R(1, 2*n+2)).
%e A060524 Row 3 polynomial R(3, x) = 5*x + x^3:
%e A060524 (3/2)^2 * Pi = 7 + 4*(3^4)*Sum_{n >= 1} (-1)^(n+1)/((2*n + 1)*R(3, 2*n)*R(3, 2*n+2)).
%e A060524 Row 5 polynomial R(5, x) = 89*x + 30*x^3 + x^5:
%e A060524 ((3*5)/(2*4))^2 * Pi = 11 + 4*(3*5)^4*Sum_{n >= 1} (-1)^(n+1)/((2*n + 1)*R(5, 2*n)*R(5, 2*n+2)). (End)
%p A060524 with(combinat):
%p A060524 b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
%p A060524       add(multinomial(n, n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1)*
%p A060524       `if`(irem(i, 2)=1, x^j, 1), j=0..n/i))))
%p A060524     end:
%p A060524 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
%p A060524 seq(T(n), n=0..12);  # _Alois P. Heinz_, Mar 09 2015
%p A060524 # alternative
%p A060524 A060524 := proc(n,k)
%p A060524     option remember;
%p A060524     if n<k or n<0 or k<0 then
%p A060524         0;
%p A060524     elif n=0 and k= 0 then
%p A060524         1;
%p A060524     else
%p A060524         procname(n-1,k-1)+(n-1)^2*procname(n-2,k) ;
%p A060524     end if;
%p A060524 end proc: # _R. J. Mathar_, Jul 06 2023
%t A060524 nn = 6; Range[0, nn]! CoefficientList[
%t A060524    Series[(1 - x^2)^(-1/2) ((1 + x)/(1 - x))^(y/2), {x, 0, nn}], {x, y}] // Grid  (* _Geoffrey Critzer_, Aug 28 2012 *)
%Y A060524 Cf. A060338, A060523, A094368, A028353 (col 1), A103916 (col 2), A103917 (col 3), A103918 (col 4).
%Y A060524 Cf. A111594 (associated Sheffer polynomials), A142979, A142983.
%K A060524 easy,nonn,tabl
%O A060524 0,8
%A A060524 _Vladeta Jovovic_, Apr 01 2001