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A060081 Exponential Riordan array (sech(x), tanh(x)). 7
1, 0, 1, -1, 0, 1, 0, -5, 0, 1, 5, 0, -14, 0, 1, 0, 61, 0, -30, 0, 1, -61, 0, 331, 0, -55, 0, 1, 0, -1385, 0, 1211, 0, -91, 0, 1, 1385, 0, -12284, 0, 3486, 0, -140, 0, 1, 0, 50521, 0, -68060, 0, 8526, 0, -204, 0, 1, -50521, 0, 663061 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Previous name was: "Triangle of coefficients (lower triangular matrix) of certain (binomial) convolution polynomials related to 1/cosh(x) and tanh(x). Use trigonometric functions for the unsigned version".

Row sums give A009265(n) (signed); A009244(n) (unsigned). Column sequences without interspersed zeros and unsigned: A000364 (Euler), A000364, A060075-8 for m=0,...,5.

a(n,m) = ((-1)^((n-m)/2))*ay(m+1,(n-m)/2) if n-m is even, else 0; where the rectangular array ay(n,m) is defined in A060058 Formula.

The row polynomials p(n,x) appear in a problem of thermo field dynamics (Bogoliubov transformation for the harmonic Bose oscillator). See the link to a .ps.gz file where they are called R_{n}(x).

The inverse of this Sheffer matrix with elements a(n,m) is the Sheffer matrix A060524. This Sheffer triangle appears in the Moyal star product of the harmonic Bose oscillator: x^{*n} = Sum_{m=0..n} a(n,m) x^m with x = 2 (bar a) a/hbar. See the Th. Spernat link, pp. 28, 29, where the unsigned version is used for y=-ix. - Wolfdieter Lang, Jul 22 2005

In the umbral calculus (see Roman reference under A048854) the p(n,x) are called Sheffer for (g(t)=1/cosh(arctanh(t)) = 1/sqrt(1-t^2), f(t)=arctanh(t)).

p(n,x) := Sum_{m=0..n} a(n,m)*x^m, n >= 0, are monic polynomials satisfying p(n,x+y) = Sum_{k=0..n} binomial(n,k)*p(k,x)*q(n-k,y) (binomial, also called exponential, convolution polynomials) with the row polynomials of the associated triangle q(n,x) := Sum_{m=0..n} A111593(n,m)*x^m. E.g.f. for p(n,x) is exp(x*tanh(z))*cosh(z)(signed). [Corrected by Wolfdieter Lang, Sep 12 2005]

Exponential Riordan array [sech(x), tanh(x)]. Unsigned triangle is [sec(x), tan(x)]. - Paul Barry, Jan 10 2011

REFERENCES

W. Lang, Two normal ordering problems and certain Sheffer polynomials, in Difference Equations, Special Functions and Orthogonal Polynomials, edts. S. Elaydi et al., World Scientific, 2007, pages 354-368. [From Wolfdieter Lang, Feb 06 2009]

LINKS

Table of n, a(n) for n=0..57.

Paul Barry, Exponential Riordan arrays and permutation enumeration,Journal of Integer Sequences, Vol. 13 (2010)

W. Lang, Thermo field dynamics, exercise 29. WS 2008/2009 (in German)

Th. Spernat, Diplomarbeit 2004 (in German) (with permission)

FORMULA

E.g.f. for column m: (((tanh(x))^m)/m!)/cosh(x), m >= 0. Use trigonometric functions for unsigned case.

a(n, m) = a(n-1, m-1)-((m+1)^2)*a(n-1, m+1); a(0, 0)=1; a(n, -1) := 0, a(n, m)=0 if n < m. Use sum of the two recursion terms for unsigned case.

a(n,k) = (Sum_{q=0..n} (C(n,q)*(((-1)^(n-q)+1)*((-1)^(q-k)+1)*Sum_{j=0..q-k}(C(j+k,k)*(j+k+1)!*2^(q-j-k-2)*(-1)^(j)*Stirling2(q+1,j+k+1))/(k+1)! - Vladimir Kruchinin, Feb 12 2019

EXAMPLE

p(3,x) = -5*x + x^3.

Exponential convolution together with A111593 for row polynomials q(n,x), case n=2: -1+(x+y)^2 = p(2,x+y) = 1*p(0,x)*q(2,y) + 2*p(1,x)*q(1,y) + 1*p(2,x)*q(0,y) = 1*1*y^2 + 2*x*y + 1*(-1+x^2)*1.

Triangle begins:

  1,

  0, 1,

  -1, 0, 1,

  0, -5, 0, 1,

  5, 0, -14, 0, 1,

  0, 61, 0, -30, 0, 1,

  -61, 0, 331, 0, -55, 0, 1,

  0, -1385, 0, 1211, 0, -91, 0, 1,

  1385, 0, -12284, 0, 3486, 0, -140, 0, 1,

  0, 50521, 0, -68060, 0, 8526, 0, -204, 0, 1,

  -50521, 0, 663061, 0, -281210, 0, 18522, 0, -285, 0, 1,

  ...

As a right-aligned triangle:

                                                       1;

                                                    0, 1;

                                                -1, 0, 1;

                                           0,   -5, 0, 1;

                                        5, 0,  -14, 0, 1;

                                 0,    61, 0,  -30, 0, 1;

                            -61, 0,   331, 0,  -55, 0, 1;

                     0,   -1385, 0,  1211, 0,  -91, 0, 1;

               1385, 0,  -12284, 0,  3486, 0, -140, 0, 1;

          0,  50521, 0,  -68060, 0,  8526, 0, -204, 0, 1;

  -50521, 0, 663061, 0, -281210, 0, 18522, 0, -285, 0, 1;

  ...

Production matrix begins

   0,   1;

  -1,   0,   1;

   0,  -4,   0,   1;

   0,   0,  -9,   0,   1;

   0,   0,   0, -16,   0,   1;

   0,   0,   0,   0, -25,   0,   1;

   0,   0,   0,   0,   0, -36,   0,   1;

   0,   0,   0,   0,   0,   0, -49,   0,   1;

   0,   0,   0,   0,   0,   0,   0, -64,   0,   1;

- Paul Barry, Jan 10 2011

MAPLE

riordan := (d, h, n, k) -> coeftayl(d*h^k, x=0, n)*n!/k!:

A060081 := (n, k) -> riordan(sech(x), tanh(x), n, k):

seq(print(seq(A060081(n, k), k=0..n)), n=0..5); # Peter Luschny, Apr 15 2015

MATHEMATICA

max = 12; t = Transpose[ Table[ PadRight[ CoefficientList[ Series[ Tanh[x]^m/m!/Cosh[x], {x, 0, max}], x], max + 1, 0]*Table[k!, {k, 0, max}], {m, 0, max}]]; Flatten[ Table[t[[n, k]], {n, 1, max}, {k, 1, n}]] (* Jean-François Alcover, Sep 29 2011 *)

PROG

(Sage)

def A060081_triangle(dim): # computes unsigned T(n, k).

    M = matrix(ZZ, dim, dim)

    for n in (0..dim-1): M[n, n] = 1

    for n in (1..dim-1):

        for k in (0..n-1):

            M[n, k] = M[n-1, k-1]+(k+1)^2*M[n-1, k+1]

    return M

A060081_triangle(9) # Peter Luschny, Sep 19 2012

CROSSREFS

Sequence in context: A230844 A054672 A021670 * A197515 A326185 A293508

Adjacent sequences:  A060078 A060079 A060080 * A060082 A060083 A060084

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang, Mar 29 2001

EXTENSIONS

New name (using a comment from Paul Barry) from Peter Luschny, Apr 15 2015

STATUS

approved

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Last modified December 2 16:17 EST 2021. Contains 349445 sequences. (Running on oeis4.)