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A060081
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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.
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6
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| 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. - W. 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(a(n,m)*x^m,m=0..n), n >= 0, are monic polynomials satisfying p(n,x+y) = sum(binomial(n,k)*p(k,x)*q(n-k,y),k=0..n) (binomial, also called exponential, convolution polynomials) with the row polynomials of the associated triangle q(n,x):= sum(A111593(n,m)*x^m,m=0..n). E.g.f. for p(n,x) is exp(x*tanh(z))*cosh(z)(signed). Corrected by W. Lang Sep 12 2005.
Exponential Riordan array [sech(x), tanh(x)]. Unsigned triangle is [sec(x), tan(x)]. [From Paul Barry (pbarry(AT}wit.ie), Jan 10 2011]
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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 (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Feb 06 2009]
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LINKS
| Thermo field dynamics exercise 29 (in German)
Th. Spernat, Diplomarbeit 2004 (in German)
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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.
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EXAMPLE
| {1}; {0,1}; {-1,0,1}; {0,-5,0,1}; ... 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, 0
-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
[From Paul Barry (pbarry(AT}wit.ie), Jan 10 2011]
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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}]] (* From Jean-François Alcover, Sep 29 2011 *)
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CROSSREFS
| Sequence in context: A174859 A054672 A021670 * A197515 A083861 A097591
Adjacent sequences: A060078 A060079 A060080 * A060082 A060083 A060084
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KEYWORD
| sign,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 29 2001
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EXTENSIONS
| In line %D, first entry: changed path from ~wl/etfth12.ps.gz to ~wl/EISpub/etfth0812.ps.gz Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jan 21 2010
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