A060524 -id:A060524 - cp's OEIS frontend

A094368 Triangle M(k,n) read by rows: coefficients of Meixner polynomials.

1, 1, -1, 1, -5, 1, -14, 9, 1, -30, 89, 1, -55, 439, -225, 1, -91, 1519, -3429, 1, -140, 4214, -24940, 11025, 1, -204, 10038, -122156, 230481, 1, -285, 21378, -463490, 2250621, -893025, 1, -385, 41778, -1467290, 14466221, -23941125, 1, -506

Offset: 1

Ralf Stephan, Jun 03 2004

			z,
z^2 - 1,
z^3 - 5*z,
z^4 - 14*z^2 + 9,
z^5 - 30*z^3 + 89*z,
z^6 - 55*z^4 + 439*z^2 - 225,
z^7 - 91*z^5 + 1519*z^3 - 3429*z,
z^8 - 140*z^6 + 4214*z^4 - 24940*z^2 + 11025,
z^9 - 204*z^7 + 10038*z^5 - 122156*z^3 + 230481*z,

Paul L. Butzer and Tom H. Koornwinder, Josef Meixner: His life and his orthogonal polynomials, Indagationes Mathematicae, Volume 30, Issue 1, January 2019, Pages 250-264.
Dominique Foata, Combinatoire des identités sur les polynomes de Meixner, Sem. Loth. de Comb. B06c (1982).
Nicolas Loizeau, Berislav Buča, and Dries Sels, Opening Krylov space to access all-time dynamics via dynamical symmetries, arXiv:2503.07403 [quant-ph], 2025. See p. 9.
Josef Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. Lond. Math. Soc. 9 (1934), 6-13.
M. Micu, Continuous Hahn polynomials, J. Math. Phys. 34 (3) (1993), 1197-1205.
Eric Weisstein's World of Mathematics, Meixner Polynomial of the Second Kind

Essentially the same as A060338.

Cf. A060524.

Recurrence: M(0, z) = 1, M(1, z) = z, M(n+1, z) = z*M(n, z) - n^2*M(n-1, z).
G.f.: exp(z*arctan(x)) / sqrt(1+x^2).
The n-th (unsigned) row polynomial R(n, x) = (-i)^n * M(n, i*x) = n!*Sum_{k = 0..n} 2^k*binomial(n, k)*binomial(x/2 - 1/2, k). - Peter Bala, Mar 10 2024

A204420 Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.

Original entry on oeis.org

1, 0, 1, 0, 6, 3, 0, 120, 90, 15, 0, 5040, 4620, 1260, 105, 0, 362880, 378000, 132300, 18900, 945, 0, 39916800, 45571680, 18711000, 3534300, 311850, 10395, 0, 6227020800, 7628100480, 3511347840, 794593800, 94594500, 5675670, 135135

Offset: 0

José H. Nieto S., Jan 15 2012

The row polynomials t(n,x):= sum(T(n,k)*x^k, k=0..n) satisfy the recurrence relation t(n,x) = (2n-1)*(x+2n-2)*t(n-1,x), with t(0,x)=1, hence t(n,x)=(2n-1)!!*x(x+2)(x+4)...(x+2n-2).

			1;
0,        1,
0,        6,        3;
0,      120,       90,       15;
0,     5040,     4620,     1260,     105;
0,   362880,   378000,   132300,   18900,    945;
0, 39916800, 45571680, 18711000, 3534300, 311850, 10395;

Alois P. Heinz, Rows n = 0..85, flattened
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
Angelo Lucia and Amanda Young, A Nonvanishing Spectral Gap for AKLT Models on Generalized Decorated Graphs, arXiv:2212.11872 [math-ph], 2022.

Cf. A049218, A060523, A060524, A132393, A048994.

Row sums give: A001818. - Alois P. Heinz, Jul 21 2013

T_row:= proc(n) local k; seq(doublefactorial(2*n-1)*2^(n-k)* coeff(expand(pochhammer(x, n)), x, k), k=0..n) end: seq(T_row(n), n=0..10);

nn=12;Prepend[Map[Prepend[Select[#,#>0&],0]&,Table[(Range[0, nn]!CoefficientList[ Series[(1-x^2)^(-y/2),{x,0,nn}],{x,y}])[[n]],{n,3,nn,2}]],{1}]//Grid (* Geoffrey Critzer, Jul 21 2013 *)

T(n,k) = (2*n)!/(2^n*n!)*(-2)^(n-k)*stirling(n,k,1); \\ Andrew Howroyd, Feb 12 2018

T(n,k) = (2n-1)!!*2^(n-k)*A132393(n,k).
T(n,k) = (2n-1)T(n-1,k-1) + (2n-1)(2n-2)*T(n-1,k); T(0,0)=1, T(n,0)=0 for n>0,
T(n,n) = (2n-1)!! = A001147(n).
T(n,1) = (2n-1)! = A009445(n-1).

A230696 Triangle read by rows related to double factorial of odd numbers (A001147).

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 11, 0, 4, 25, 0, 72, 0, 8, 0, 589, 0, 340, 0, 16, 2025, 0, 7010, 0, 1328, 0, 32, 0, 75319, 0, 55160, 0, 4592, 0, 64, 342225, 0, 1335328, 0, 334752, 0, 14592, 0, 128

Offset: 0

Philippe Deléham, Oct 27 2013

The row polynomials t(n,x):= sum(T(n,k)*x^k, k=0..n) satisfy the recurrence relation t(n,x) = 2x*t(n-1,x) + ((2n-3)^2)*t(n-2,x); t(0,x) = 1, t(1,x) = x.

			Triangle begins:
1
0, 1
1, 0, 2
0, 11, 0, 4
25, 0, 72, 0, 8
0, 589, 0, 340, 0, 16
2025, 0, 7010, 0, 1328, 0, 32
0, 75319, 0, 55160, 0, 4592, 0, 64

T(2n,0) = A007696(n)^2.
T(n,n) = A011782(n).
A001147 (row sums).
Cf. A060524 (similar sequence).

T(n,k) = 2*T(n-1,k-1) + ((2n-3)^2)*T(n-2,k); T(0,0) = 1, T(1,0) = 0, T(1,1) = 1, T(n,k) = 0 if k>n or if k<0.

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A094368 Triangle M(k,n) read by rows: coefficients of Meixner polynomials.

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A204420 Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.

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A230696 Triangle read by rows related to double factorial of odd numbers (A001147).

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