

A218272


Infinitesimal generator for transpose of the Pascal matrix A007318 (as upper triangular matrices).


6



0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0
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OFFSET

0,5


COMMENTS

Matrix T begins
0,1;
0,0,2;
0,0,0,3;
0,0,0,0,4;
0,0,0,0,0,5;
0,0,0,0,0,0,6;
T is the transpose of A132440.
Let M(t) = exp(t*T) = limit [1 + t*T/n]^n as n tends to infinity.
Then M(1) = the transpose of the lower triangular Pascal matrix A007318, with inverse M(1).
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_(n1)(x) and
R P_n(x) = P_(n+1)(x), the matrix T represents the action of L 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).
See A132440 as an analog and more general discussion.
Sum(n>=0, c_n T^n / n!) = e^(c.T) gives the MaurerCartan form matrix for the one dimensional Leibniz group defined by multiplication of a Taylor series by the formal Taylor series e^(c.x) (cf. Olver).  Tom Copeland, Nov 05 2015


LINKS

Table of n, a(n) for n=0..79.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
P. Olver, The canonical contact form pg. 8


FORMULA

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(n) = (n+1) * a(n+1),
2) B(x) = D A(x), or
3) EB(x) = DxD EA(x),
where D is the derivative w.r.t. x.
So the exponentiated operator can be characterized as
4) exp(t*T) A(x) = exp(t*D) A(x) = A(x+t),
5) exp(t*T) EA(x) = exp(t*DxD) EA(x) = exp[x*a/(1+t*a)]/(1+t*a),
= sum(n=0 to infn) (1+t*a)^(n1) (x*a)^n/n!, where umbrally
a^n *(1+t*a)^(n1)= sum(j=0 to infn) binom(n+j,j)a(n+j)t^j,
6) exp(t*T) EA(x) = sum(n=0 to infn) a(n) t^n Lag(n,x/t),
where Lag(n,x) are the Laguerre polynomials (A021009), or
7) [exp(t*T) * a]_n = [M(t) * a]_n
= sum(j=0 to infn)binom(n+j,j)a(n+j)t^j.
For more on the operator DxD, see A021009 and references in A132440.


MATHEMATICA

Table[PadLeft[{n+1}, n+2], {n, 0, 11}] // Flatten (* JeanFrançois Alcover, Apr 30 2014 *)


CROSSREFS

Cf. A134402.
Sequence in context: A232747 A130460 A132440 * A134402 A174712 A127647
Adjacent sequences: A218269 A218270 A218271 * A218273 A218274 A218275


KEYWORD

nonn,easy,tabf


AUTHOR

Tom Copeland, Oct 24 2012


STATUS

approved



