A122850 Exponential Riordan array (1, sqrt(1+2x)-1).

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

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..54.

P. Bala, The white diamond product of power series

Orli Herscovici, Study of the p,q-deformed Touchard polynomials, arXiv:1904.07674 [math.CO], 2019.

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)

Sum_{k=0..n} (-1)^(n+k) * T(n,k) = A144301(n). - Alois P. Heinz, Aug 31 2022

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, A144301.

Sequence in context: A265608 A184962 A264436 * A132062 A065547 A143333

Adjacent sequences: A122847 A122848 A122849 * A122851 A122852 A122853

Keyword

sign,tabl,easy

Author

Paul Barry, Sep 14 2006

Extensions

More terms from Alois P. Heinz, Aug 31 2022

Status

approved