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.

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 A127557

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