A122850 -id:A122850 - cp's OEIS frontend

A001497 Triangle of coefficients of Bessel polynomials (exponents in decreasing order).

1, 1, 1, 3, 3, 1, 15, 15, 6, 1, 105, 105, 45, 10, 1, 945, 945, 420, 105, 15, 1, 10395, 10395, 4725, 1260, 210, 21, 1, 135135, 135135, 62370, 17325, 3150, 378, 28, 1, 2027025, 2027025, 945945, 270270, 51975, 6930, 630, 36, 1, 34459425, 34459425, 16216200, 4729725, 945945, 135135, 13860, 990, 45, 1

Offset: 0

N. J. A. Sloane

The (reverse) Bessel polynomials P(n,x):=Sum_{m=0..n} a(n,m)*x^m, the row polynomials, called Theta_n(x) in the Grosswald reference, solve x*(d^2/dx^2)P(n,x) - 2*(x+n)*(d/dx)P(n,x) + 2*n*P(n,x) = 0.
With the related Sheffer associated polynomials defined by Carlitz as
B(0,x) = 1
B(1,x) = x
B(2,x) = x + x^2
B(3,x) = 3 x + 3 x^2 + x^3
B(4,x) = 15 x + 15 x^2 + 6 x^3 + x^4
... (see Mathworld reference), then P(n,x) = 2^n * B(n,x/2) are the Sheffer polynomials described in A119274. - Tom Copeland, Feb 10 2008
Exponential Riordan array [1/sqrt(1-2x), 1-sqrt(1-2x)]. - Paul Barry, Jul 27 2010
From Vladimir Kruchinin, Mar 18 2011: (Start)
For B(n,k){...} the Bell polynomial of the second kind we have
B(n,k){f', f'', f''', ...} = T(n-1,k-1)*(1-2*x)^(k/2-n), where f(x) = 1-sqrt(1-2*x).
The expansions of the first few rows are:
1/sqrt(1-2*x);
1/(1-2*x)^(3/2), 1/(1-2*x);
3/(1-2*x)^(5/2), 3/(1-2*x)^2, 1/(1-2*x)^(3/2);
15/(1-2*x)^(7/2), 15/(1-2*x)^3, 6/(1-2*x)^(5/2), 1/(1-2*x)^2. (End)
Also the Bell transform of A001147 (whithout column 0 which is 1,0,0,...). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016
Antidiagonals of A099174 are rows of this entry. Dividing each diagonal by its first element generates A054142. - Tom Copeland, Oct 04 2016
The row polynomials p_n(x) of A107102 are (-1)^n B_n(1-x), where B_n(x) are the modified Carlitz-Bessel polynomials above, e.g., (-1)^2 B_2(1-x) = (1-x) + (1-x)^2 = 2 - 3 x + x^2 = p_2(x). - Tom Copeland, Oct 10 2016
a(n-1,m-1) counts rooted unordered binary forests with n labeled leaves and m roots. - David desJardins, Feb 23 2019
From Jianing Song, Nov 29 2021: (Start)
The polynomials P_n(x) = Sum_{k=0..n} T(n,k)*x^k satisfy: P_n(x) - (d/dx)P_n(x) = x*P_{n-1}(x) for n >= 1.
{P(n,x)} are related to the Fourier transform of 1/(1+x^2)^(n+1) and x/(1+x^2)^(n+2):
(i) For n >= 0, real number t, we have Integral_{x=-oo..oo} exp(-i*t*x)/(1+x^2)^(n+1) dx = Pi/(2^n*n!) * P_n(|t|) * exp(-|t|);
(ii) For n >= 0, real number t, we have Integral_{x=-oo..oo} x*exp(-i*t*x)/(1+x^2)^(n+2) dx = Pi/(2^(n+1)*(n+1)!) * ((-t)*P_n(-|t|)) * exp(-|t|). (End)
Suppose that f(x) is an n-times differentiable function defined on (a,b) for 0 <= a < b <= +oo, then for n >= 1, the n-th derivative of f(sqrt(x)) on (a^2,b^2) is Sum_{k=1..n} ((-1)^(n-k)*T(n-1,k-1)*f^(k)(sqrt(x))) / (2^n*x^(n-(k/2))), where f^(k) is the k-th derivative of f. - Jianing Song, Nov 30 2023

			Triangle begins
        1,
        1,       1,
        3,       3,      1,
       15,      15,      6,      1,
      105,     105,     45,     10,     1,
      945,     945,    420,    105,    15,    1,
    10395,   10395,   4725,   1260,   210,   21,   1,
   135135,  135135,  62370,  17325,  3150,  378,  28,  1,
  2027025, 2027025, 945945, 270270, 51975, 6930, 630, 36, 1
Production matrix begins
       1,      1,
       2,      2,      1,
       6,      6,      3,     1,
      24,     24,     12,     4,     1,
     120,    120,     60,    20,     5,    1,
     720,    720,    360,   120,    30,    6,   1,
    5040,   5040,   2520,   840,   210,   42,   7,  1,
   40320,  40320,  20160,  6720,  1680,  336,  56,  8, 1,
  362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1
This is the exponential Riordan array A094587, or [1/(1-x),x], beheaded.
- _Paul Barry_, Mar 18 2011

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

T. D. Noe, Rows n = 0..50 of triangle, flattened
Peter Bala, Generalized Dobinski formulas
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, chapter 8.
Tom Copeland, A Class of Differential Operators and the Stirling Numbers
E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
O. Frink and H. L. Krall, A new class of orthogonal polynomials, Trans. Amer. Math. Soc. 65,100-115, 1945. [From _Roger L. Bagula_, Feb 15 2009]
E. Grosswald, Bessel Polynomials, Lecture Notes Math. vol. 698 1978 p. 18.
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) #09.8.3.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
B. Leclerc, Powers of staircase Schur functions and symmetric analogues of Bessel polynomials, Discrete Math., 153 (1996), 213-227.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.
Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From _N. J. A. Sloane_, Sep 15 2012
W. Mlotkowski and A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.
Feng Qi and Bai-Ni Guo, "Some Properties and Generalizations of the Catalan, Fuss, and Fuss-Catalan Numbers", Mathematical Analysis and Applications : Selected Topics (2018), Wiley, Ch. 5, 101-133.
Feng Qi, X.-T. Shi and F.-F. Liu, Several formulas for special values of the Bell polynomials of the second kind and applications, Preprint 2015.
Alexander Stoimenow, On the number of chord diagrams, Discr. Math. 218 (2000), 209-233. Lemma 2.2.
Eric Weisstein's World of Mathematics, Bessel Polynomial
Index entries for sequences related to Bessel functions or polynomials

Reflected version of A001498 which is considered the main entry.
Other versions of this same triangle are given in A144299, A111924 and A100861.
Row sums give A001515. a(n, 0)= A001147(n) (double factorials).
Cf. A104556 (matrix inverse). A039683, A122850.
Cf. A245066 (central terms).
Cf. A054142, A099174, A107102.

a001497 n k = a001497_tabl !! n !! k
a001497_row n = a001497_tabl !! n
a001497_tabl = [1] : f [1] 1 where
   f xs z = ys : f ys (z + 2) where
     ys = zipWith (+) ([0] ++ xs) (zipWith (*) [z, z-1 ..] (xs ++ [0]))
-- Reinhard Zumkeller, Jul 11 2014

/* As triangle */ [[Factorial(2*n-k)/(Factorial(k)*Factorial(n-k)*2^(n-k)): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 12 2015

f := proc(n) option remember; if n <=1 then (1+x)^n else expand((2*n-1)*x*f(n-1)+f(n-2)); fi; end;
row := n -> seq(coeff(f(n), x, n - k), k = 0..n): seq(row(n), n = 0..9);

m = 9; Flatten[ Table[(n + k)!/(2^k*k!*(n - k)!), {n, 0, m}, {k, n, 0, -1}]] (* Jean-François Alcover, Sep 20 2011 *)
y[n_, x_] := Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n-1/2, 1/x]; t[n_, k_] := Coefficient[y[n, x], x, k]; Table[t[n, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 01 2013 *)

T(k, n) = if(n>k||k<0||n<0,0,(2*k-n)!/(n!*(k-n)!*2^(k-n))) /* Ralf Stephan */

{T(n, k) = if( k<0 || k>n, 0, binomial(n, k)*(2*n-k)!/2^(n-k)/n!)}; /* Michael Somos, Oct 03 2006 */

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: A001147(n), 9) # Peter Luschny, Jan 19 2016

a(n, m) = (2*n-m)!/(m!*(n-m)!*2^(n-m)) if n >= m >= 0 else 0 (from Grosswald, p. 7).
a(n, m)= 0, n= m >= 0 (from Grosswald p. 23, (19)).
E.g.f. for m-th column: ((1-sqrt(1-2*x))^m)/(m!*sqrt(1-2*x)).
G.f.: 1/(1-xy-x/(1-xy-2x/(1-xy-3x/(1-xy-4x/(1-.... (continued fraction). - Paul Barry, Jan 29 2009
T(n,k) = if(k<=n, C(2n-k,2(n-k))*(2(n-k)-1)!!,0) = if(k<=n, C(2n-k,2(n-k))*A001147(n-k),0). - Paul Barry, Mar 18 2011
Row polynomials for n>=1 are given by 1/t*D^n(exp(x*t)) evaluated at x = 0, where D is the operator 1/(1-x)*d/dx. - Peter Bala, Nov 25 2011
The matrix product A039683*A008277 gives a signed version of this triangle. Dobinski-type formula for the row polynomials: R(n,x) = (-1)^n*exp(x)*Sum_{k = 0..inf} k*(k-2)*(k-4)*...*(k-2*(n-1))*(-x)^k/k!. Cf. A122850. - Peter Bala, Jun 23 2014

A122848 Exponential Riordan array (1, x(1+x/2)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 0, 3, 6, 1, 0, 0, 0, 15, 10, 1, 0, 0, 0, 15, 45, 15, 1, 0, 0, 0, 0, 105, 105, 21, 1, 0, 0, 0, 0, 105, 420, 210, 28, 1, 0, 0, 0, 0, 0, 945, 1260, 378, 36, 1, 0, 0, 0, 0, 0, 945, 4725, 3150, 630, 45, 1, 0, 0, 0, 0, 0, 0, 10395, 17325, 6930, 990, 55, 1, 0, 0

Offset: 0

Paul Barry, Sep 14 2006

Entries are Bessel polynomial coefficients. Row sums are A000085. Diagonal sums are A122849. Inverse is A122850. Product of A007318 and A122848 gives A100862.
T(n,k) is the number of self-inverse permutations of {1,2,...,n} having exactly k cycles. - Geoffrey Critzer, May 08 2012
Bessel numbers of the second kind. For relations to the Hermite polynomials and the Catalan (A033184 and A009766) and Fibonacci (A011973, A098925, and A092865) matrices, see Yang and Qiao. - Tom Copeland, Dec 18 2013.
Also the inverse Bell transform of the double factorial of odd numbers Product_{k= 0..n-1} (2*k+1) (A001147). For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015

			Triangle begins:
    1
    0    1
    0    1    1
    0    0    3    1
    0    0    3    6    1
    0    0    0   15   10    1
    0    0    0   15   45   15    1
    0    0    0    0  105  105   21    1
    0    0    0    0  105  420  210   28    1
    0    0    0    0    0  945 1260  378   36    1
From _Gus Wiseman_, Jan 12 2021: (Start)
As noted above, a(n) is the number of set partitions of {1..n} into k singletons or pairs. This is also the number of set partitions of subsets of {1..n} into n - k pairs. In the first case, row n = 5 counts the following set partitions:
  {{1},{2,3},{4,5}}  {{1},{2},{3},{4,5}}  {{1},{2},{3},{4},{5}}
  {{1,2},{3},{4,5}}  {{1},{2},{3,4},{5}}
  {{1,2},{3,4},{5}}  {{1},{2,3},{4},{5}}
  {{1,2},{3,5},{4}}  {{1,2},{3},{4},{5}}
  {{1},{2,4},{3,5}}  {{1},{2},{3,5},{4}}
  {{1},{2,5},{3,4}}  {{1},{2,4},{3},{5}}
  {{1,3},{2},{4,5}}  {{1},{2,5},{3},{4}}
  {{1,3},{2,4},{5}}  {{1,3},{2},{4},{5}}
  {{1,3},{2,5},{4}}  {{1,4},{2},{3},{5}}
  {{1,4},{2},{3,5}}  {{1,5},{2},{3},{4}}
  {{1,4},{2,3},{5}}
  {{1,4},{2,5},{3}}
  {{1,5},{2},{3,4}}
  {{1,5},{2,3},{4}}
  {{1,5},{2,4},{3}}
In the second case, we have:
  {{1,2},{3,4}}  {{1,2}}  {}
  {{1,2},{3,5}}  {{1,3}}
  {{1,2},{4,5}}  {{1,4}}
  {{1,3},{2,4}}  {{1,5}}
  {{1,3},{2,5}}  {{2,3}}
  {{1,3},{4,5}}  {{2,4}}
  {{1,4},{2,3}}  {{2,5}}
  {{1,4},{2,5}}  {{3,4}}
  {{1,4},{3,5}}  {{3,5}}
  {{1,5},{2,3}}  {{4,5}}
  {{1,5},{2,4}}
  {{1,5},{3,4}}
  {{2,3},{4,5}}
  {{2,4},{3,5}}
  {{2,5},{3,4}}
(End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Bala, Generalized Dobinski formulas
Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
H. Han and S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, Eur. J. Combinat. 29 (7) (2008) 1544-1554. [From _R. J. Mathar_, Mar 20 2009]
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 18.
S. Yang and Z. Qiao, The Bessel Numbers and Bessel Matrices, Journal of Mathematical Research & Exposition, July, 2011, Vol. 31, No. 4, pp. 627-636. [From _Tom Copeland_, Dec 18 2013]

Cf. A001497, A008275, A039755, A096713, A104556, A113278, A130757.
Row sums are A000085.
Column sums are A001515.
Same as A049403 but with a first column k = 0.
The same set partitions counted by number of pairs are A100861.
Reversing rows gives A111924 (without column k = 0).
A047884 counts standard Young tableaux by size and greatest row length.
A238123 counts standard Young tableaux by size and least row length.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
Cf. A000110, A000258, A320732, A321729, A339741, A339887.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n<2,1,0), 9); # Peter Luschny, Jan 27 2016

t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten
(* Second program: *)
rows = 12;
t = Join[{1, 1}, Table[0, rows]];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 23 2018,after Peter Luschny *)
sbs[{}]:={{}};sbs[set:{i_,_}]:=Join@@Function[s,(Prepend[#1,s]&)/@sbs[Complement[set,s]]]/@Cases[Subsets[set],{i}|{i,_}];
Table[Length[Select[sbs[Range[n]],Length[#]==k&]],{n,0,6},{k,0,n}] (* Gus Wiseman, Jan 12 2021 *)

{T(n,k)=if(2*kn, 0, n!/(2*k-n)!/(n-k)!*2^(k-n))} /* Michael Somos, Oct 03 2006 */

# uses[inverse_bell_transform from A265605]
multifact_2_1 = lambda n: prod(2*k + 1 for k in (0..n-1))
inverse_bell_matrix(multifact_2_1, 9) # Peter Luschny, Dec 31 2015

Number triangle T(n,k) = k!*C(n,k)/((2k-n)!*2^(n-k)).
T(n,k) = A001498(k,n-k). - Michael Somos, Oct 03 2006
E.g.f.: exp(y(x+x^2/2)). - Geoffrey Critzer, May 08 2012
Triangle equals the matrix product A008275*A039755. Equivalently, the n-th row polynomial R(n,x) is given by the Type B Dobinski formula R(n,x) = exp(-x/2)*Sum_{k>=0} P(n,2*k+1)*(x/2)^k/k!, where P(n,x) = x*(x-1)*...*(x-n+1) denotes the falling factorial polynomial. Cf. A113278. - Peter Bala, Jun 23 2014
From Daniel Checa, Aug 28 2022: (Start)
E.g.f. for the m-th column: (x^2/2+x)^m/m!.
T(n,k) = T(n-1,k-1) + (n-1)*T(n-2,k-1) for n>1 and k=1..n, T(0,0) = 1. (End)

A132062 Sheffer triangle (1,1-sqrt(1-2*x)). Extended Bessel triangle A001497.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang Sep 14 2007

This is a Jabotinsky type exponential convolution triangle related to A001147 (double factorials). For Jabotinsky type triangles See the D. E. Knuth reference given under A039692.
The subtriangle (n>=m>=1) is A001497(n,m) (Bessel).
For the combinatorial interpretation in terms of unordered forests of increasing plane trees see the W. Lang comment and example under A001497.
This is a special type of Sheffer triangle. See the S. Roman reference given under A048854 (the notation here differs).
This triangle (or the A001497 subtriangle) appears as generalized Stirling numbers of the second kind, S2p(-1,n,m):=S2(-k;m,m)*(-1)^(n-m) for k=1, eqs. (27)-(29) of the W. Lang reference.
Also the Bell transform of the double factorial of odd numbers A001147. For the Bell transform of the double factorial of even numbers A000165 see A039683. For the definition of the Bell transform see A264428. - Peter Luschny, Dec 20 2015

			[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]

Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From N. J. A. Sloane, Sep 15 2012
Steven Roman, The Umbral Calculus, Pure and Applied Mathematics, 111, Academic Press, 1984. (p. 78) [Emanuele Munarini, Oct 10 2017]

Leonard Carlitz, A Note on the Bessel Polynomials, Duke Math. J. 24 (2) (1957), 151-162. [_Emanuele Munarini_, Oct 10 2017]
H. Han and S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, Eur. J. Combinat. 29 (7) (2008) 1544-1554. [From _R. J. Mathar_, Mar 20 2009]
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First 10 rows.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 18.

Columns m=1: A001147.
Row sums give [1, A001515]. Alternating row sums give [1, -A000806].
Cf. A122850. - R. J. Mathar, Mar 20 2009
Cf. A039683, A264428.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> doublefactorial(2*n-1), 9); # Peter Luschny, Jan 27 2016
# Alternative:
egf := exp(y*(1 - sqrt(1 - 2*x))): serx := series(egf, x, 12):
coefx := n -> n!*coeff(serx, x, n): row := n -> seq(coeff(coefx(n), y, k), k = 0..n): for n from 0 to 8 do row(n) od;  # Peter Luschny, Apr 25 2024

Table[If[k <= n, Binomial[2n-2k,n-k] Binomial[2n-k-1,k-1] (n-k)!/2^(n-k), 0], {n, 0, 6}, {k, 0, n}] // Flatten (* Emanuele Munarini, Oct 10 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[(2#-1)!!&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

# uses[bell_transform from A264428]
def A132062_row(n):
    a = sloane.A001147
    dblfact = a.list(n)
    return bell_transform(n, dblfact)
[A132062_row(n) for n in (0..9)] # Peter Luschny, Dec 20 2015

a(n,m)=0 if n

E.g.f. m-th column ((x*f2p(1;x))^m)/m!, m>=0. with f2p(1;x):=1-sqrt(1-2*x)= x*c(x/2) with the o.g.f.of A000108 (Catalan).

From Emanuele Munarini, Oct 10 2017: (Start)

a(n,k) = binomial(2*n-2*k,n-k)*binomial(2*n-k-1,k-1)*(n-k)!/2^(n-k).

The row polynomials p_n(x) (studied by Carlitz) satisfy the recurrence: p_{n+2}(x) - (2*n+1)*p_{n+1}(x) - x^2*p_n(x) = 0. (End)

T(n, k) = n! [y^k] [x^n] exp(y*(1 - sqrt(1 - 2*x))). - Peter Luschny, Apr 25 2024

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A001497 Triangle of coefficients of Bessel polynomials (exponents in decreasing order).

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A122848 Exponential Riordan array (1, x(1+x/2)).

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A132062 Sheffer triangle (1,1-sqrt(1-2*x)). Extended Bessel triangle A001497.

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