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
Examples
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 ...
Links
- 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.
Programs
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Maple
# The function BellMatrix is defined in A264428. BellMatrix(n -> (-1)^n*doublefactorial(2*n-1), 9); # Peter Luschny, Jan 27 2016
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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 *) -
Sage
# uses[bell_matrix from A264428] bell_matrix(lambda n: 1 if n<2 else 0, 12).inverse() # Peter Luschny, Jan 19 2016
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
Extensions
More terms from Alois P. Heinz, Aug 31 2022
Comments