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
Also called Bessel numbers of first kind, and denoted b(n,k). - Abdelhay Benmoussa, Aug 30 2025
LINKS
Peter Bala, The white diamond product of power series
Hyuk Han and Seunghyun Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, arXiv:math/0406378 [math.CO], 2004.
Orli Herscovici, Study of the p,q-deformed Touchard polynomials, arXiv:1904.07674 [math.CO], 2019.
Milan Janjić, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Wikipedia, Bessel polynomials
Simon Willerton, The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials, arXiv:1708.03227 [math.MG], 2017-2020.
Shengliang Yang and Zhanke Qiao, The Bessel Numbers and Bessel Matrices, Journal of Mathematical Research and Exposition, Jul. 2011, Vol. 31, No. 4, pp. 627-636.
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
From Abdelhay Benmoussa, Sep 06 2025: (Start)
T(n,k) = [x^k] e^(-x)*[ (xI)^n(e^t)(x) + Sum_{k=0..n-1} (-1)^(n-1-k) (2(n-1-k)-1)!!*x^(2k+1)/(2k)!! ], where I(f)(x) = Integral_{0..x} f(t)dt.
(x^(-1) D)^n = Sum_{k=0..n} T(n,k)*x^(k-2n)*D^k, where D=d/dx.
(xI)^n = Sum_{k=1..n} T(n,k)*x^k*I^{2n-k}, where I(f)(x) = Integral_{a..x} f(t) dt.
T(n,k) = (-1)^(n-k)*A001498(n-1,n-k).
T(n,k) = (-1)^(n-k)*(2n-k-1)!/2^(n-k)*(n-k)!*(k-1)! if 1=<k=<n and 0 if 1=<n<k. (End)
T(n,k) = (1/k!) * Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * Product_{j=0..n-1}(i-2j). - Abdelhay Benmoussa, Oct 02 2025
x*(x-2)*...*(x-2*(n-1)) = Sum_{k=0..n} T(n,k)*x*(x-1)*...*(x-k+1). - Abdelhay Benmoussa, Jan 05 2026
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
(SageMath) # uses[bell_matrix from A264428]
bell_matrix(lambda n: 1 if n<2 else 0, 12).inverse() # Peter Luschny, Jan 19 2016
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Sep 14 2006
EXTENSIONS
More terms from Alois P. Heinz, Aug 31 2022
STATUS
approved