1,1

LetA129184= matrix M, then M*V, (V a vector); shifts V to the right, preceded by zeros. Example: M*V, V = [1, 2, 3,...] = [0, 1, 2, 3,...].A129185= left shift operator.

Given a polynomial sequence p_n(x) with p_0(x)=1 and the lowering and raising operators L and R defined by L P_n(x)= n * P_(n-1)(x) and R P_n(x)= P_(n+1)(x), the matrix T represents the action of R in the p_n(x) basis. For p_n(x) = x^n, L = D = d/dx and R = x. For p_n(x)= x^n/n!, L= DxD and R=D^(-1). -Tom Copeland,Nov 10 2012

Andrew Howroyd,Table of n, a(n) for n = 1..1275

Infinite lower triangular matrix with all 1's in the subdiagonal and the rest zeros.

FromTom Copeland,Nov 10 2012: (Start)

Let M(t) = I/(I-t*T) = I + t*T + (t*T)^2 +... where T is the shift operator matrix and I the Identity matrix. Then the inverse matrix is MI(t)=(I-tT) and M(t) isA000012with each n-th diagonal multiplied by t^n. M(1)=A000012with inverse MI(1)=A167374.Row sums of M(2), M(3), and M(4) areA000225,A003462,andA002450.

Let E(t)=exp(t*T) with inverse E(-t). Then E(t) isA000012with each n-th diagonal multiplied by t^n/n! and each row represents e^t truncated at the n+1 term.

The matrix operation b = T*a can be characterized in several ways in terms of the coefficients a(n) and b(n), their o.g.f.s A(x) and B(x), or e.g.f.s EA(x) and EB(x):

1) b(0) = 0, b(n) = a(n-1),

2) B(x) = x A(x), or

3) EB(x) = D^(-1) EA(x), where D^(-1)x^j/j! = x^(j+1)/(j+1)!.

The operator M(t) can be characterized as

4)M(t)EA(x)= sum(n>=0)a(n)[e^(x*t)-[1+x*t+...+ (x*t)^(n-1)/(n-1)!]]/t^n

= exp(a*D_y)[t*e^(x*t)-y*e(x*y)]/(t-y) <evaluated at y=0>

= [t*e^(x*t)-a*e(x*a)]/(t-a), umbrally where (a)^k=a_k,

5)[M(t) * a]_n = a(0)t^n +a(1)t^(n-1)+a(2)t^(n-2)+...+a(n).

The exponentiated operator can be characterized as

6) E(t) A(x) = exp(t*x) A(x),

7) E(t) EA(x) = exp(t*D^(-1)) EA(x)

8) [E(t) * a]_n = a(0)t^n/n! + a(1)t^(n-1)/(n-1)! +... + a(n).

(End)

a(n) =A010054(n+1). -Andrew Howroyd,Feb 02 2020

First few rows of the triangle:

0;

1, 0;

0, 1, 0;

0, 0, 1, 0;

0, 0, 0, 1, 0;

...

Cf.A129185,A129186.

Cf.A010054.

Sequence in context:A068716A188020A179828*A129185A283683A118605

Adjacent sequences:A129181A129182A129183*A129185A129186A129187

nonn,tabl,less,easy

Gary W. Adamson,Apr 01 2007

Terms a(46) and beyond fromAndrew Howroyd,Feb 02 2020

approved