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A122850 Exponential Riordan array (1, sqrt(1+2x)-1). 3
1, 0, 1, 0, -1, 1, 0, 3, -3, 1, 0, -15, 15, -6, 1, 0, 105, -105, 45, -10, 1, 0, -945, 945, -420, 105, -15, 1, 0, 10395, -10395, 4725, -1260, 210, -21, 1, 0, -135135, 135135, -62370, 17325, -3150 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Inverse of number triangle A122848. Entries are Bessel polynomial coefficients. Row sums are A000806.

Also the inverse Bell transform of the sequence "g(n) = 1 if n<2 else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

LINKS

Table of n, a(n) for n=0..41.

P. Bala, The white diamond product of power series

M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3

Wikipedia, Bessel polynomials

S. Willerton, The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials, arXiv:1708.03227v1 [math.MG], 2017.

FORMULA

T(n,k) = (-1)^(n-k)*A132062(n,k). - Philippe Deléham, Nov 06 2011

Triangle equals the matrix product A039757*A008277. Dobinski-type formula for the row polynomials: R(n,x) = x*exp(-x)*Sum_{k = 0..inf} (k-1)*(k-3)*(k-5)*...*(k-(2*n-3))*x^k/k! for n >= 1. Cf. A001497. - Peter Bala, Jun 23 2014

From Peter Bala, Jan 09 2018: (Start)

Alternative Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k = 0..inf} k*(k-2)*(k-4)*...*(k-(2*n-2))*x^k/k!.

Equivalently, R(n,x) = x o (x-2) o (x-4) o...o (x-(2*n-2)), where o denotes the white diamond product of polynomials. See the Bala link for the definition and details.

The white diamond products (x-1) o (x-3) o...o (x-(2*n-3)) give the row polynomials of the array with a factor of x removed.

If d is the first derivative operator f -> d/dx(f(x)) and D is the operator f(x) -> 1/x*d/dx(f(x)) then x^(2*n)*D^n = R(n,x*d), with the understanding that (x*d)^k is to interpreted as the operator f(x) -> x^k*d^k(f(x))/dx^k. (End)

EXAMPLE

Triangle begins

1

0 1

0 -1 1

0 3 -3 1

0 -15 15 -6 1

0 105 -105 45 -10 1

0 -945 945 -420 105 -15 1

0 10395 -10395 4725 -1260 210 -21 1

0 -135135 135135 -62370 17325 -3150 378 -28 1

0 2027025 -2027025 945945 -270270 51975 -6930 630 -36 1

0 -34459425 34459425 -16216200 4729725 -945945 135135 -13860 990 -45 1

MAPLE

# The function BellMatrix is defined in A264428.

BellMatrix(n -> (-1)^n*doublefactorial(2*n-1), 9); # Peter Luschny, Jan 27 2016

MATHEMATICA

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

rows = 12;

M = BellMatrix[Function[n, (-1)^n (2n-1)!!], rows];

Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 26 2018, after Peter Luschny *)

PROG

(Sage) # uses[bell_matrix from A264428]

bell_matrix(lambda n: 1 if n<2 else 0, 12).inverse() # Peter Luschny, Jan 19 2016

CROSSREFS

Cf. A000806, A001497, A008277, A039757, A122848, A132062.

Sequence in context: A265608 A184962 A264436 * A132062 A065547 A143333

Adjacent sequences:  A122847 A122848 A122849 * A122851 A122852 A122853

KEYWORD

easy,sign,tabl

AUTHOR

Paul Barry, Sep 14 2006

STATUS

approved

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Last modified August 11 23:04 EDT 2020. Contains 336434 sequences. (Running on oeis4.)