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A060081
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Exponential Riordan array (sech(x), tanh(x)).
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7
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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
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OFFSET
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0,8
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COMMENTS
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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".
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
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REFERENCES
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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]
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LINKS
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FORMULA
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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
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EXAMPLE
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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;
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MAPLE
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riordan := (d, h, n, k) -> coeftayl(d*h^k, x=0, n)*n!/k!:
A060081 := (n, k) -> riordan(sech(x), tanh(x), n, k):
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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