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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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_(n-1)(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 Maurer-Cartan 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 p. 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} (1+t*a)^(-n-1) (x*a)^n/n!, where umbrally

     a^n *(1+t*a)^(-n-1) = Sum_{j>0} binomial(n+j,j)a(n+j)t^j,

  6) exp(t*T) EA(x) = Sum_{n>=0} 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} binomial(n+j,j)a(n+j)t^j.

For more on the operator DxD, see A021009 and references in A132440.

EXAMPLE

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;

  ...

MATHEMATICA

Table[PadLeft[{n+1}, n+2], {n, 0, 11}] // Flatten (* Jean-Fran├žois 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

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Last modified May 20 18:46 EDT 2018. Contains 304347 sequences. (Running on oeis4.)