0,8

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.

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]

Thermo field dynamics exercise 29 (in German)

Th. Spernat, Diplomarbeit 2004 (in German)

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.

{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.

Sequence in context: A117015 A054672 A021670 this_sequence A083861 A097591 A164652

Adjacent sequences: A060078 A060079 A060080 this_sequence A060082 A060083 A060084

sign,easy,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 29 2001