A001515 -id:A001515 - cp's OEIS frontend

A001498 Triangle a(n,k) (n >= 0, 0 <= k <= n) of coefficients of Bessel polynomials y_n(x) (exponents in increasing order).

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

Offset: 0

N. J. A. Sloane

The row polynomials with exponents in increasing order (e.g., third row: 1+3x+3x^2) are Grosswald's y_{n}(x) polynomials, p. 18, Eq. (7).
Also called Bessel numbers of first kind.
The triangle a(n,k) has factorization [C(n,k)][C(k,n-k)]Diag((2n-1)!!) The triangle a(n-k,k) is A100861, which gives coefficients of scaled Hermite polynomials. - Paul Barry, May 21 2005
Related to k-matchings of the complete graph K_n by a(n,k)=A100861(n+k,k). Related to the Morgan-Voyce polynomials by a(n,k)=(2k-1)!!*A085478(n,k). - Paul Barry, Aug 17 2005
Related to Hermite polynomials by a(n,k)=(-1)^k*A060821(n+k, n-k)/2^n. - Paul Barry, Aug 28 2005
The row polynomials, the Bessel polynomials y(n,x):=Sum_{m=0..n} (a(n,m)*x^m) (called y_{n}(x) in the Grosswald reference) satisfy (x^2)*(d^2/dx^2)y(n,x) + 2*(x+1)*(d/dx)y(n,x) - n*(n+1)*y(n,x) = 0.
a(n-1, m-1), n >= m >= 1, enumerates unordered n-vertex forests composed of m plane (aka ordered) increasing (rooted) trees. Proof from the e.g.f. of the first column Y(z):=1-sqrt(1-2*z) (offset 1) and the Bergeron et al. eq. (8) Y'(z)= phi(Y(z)), Y(0)=0, with out-degree o.g.f. phi(w)=1/(1-w). See their remark on p. 28 on plane recursive trees. For m=1 see the D. Callan comment on A001147 from Oct 26 2006. - Wolfdieter Lang, Sep 14 2007
The asymptotic expansions of the higher order exponential integrals E(x,m,n), see A163931 for information, lead to the Bessel numbers of the first kind in an intriguing way. For the first four values of m these asymptotic expansions lead to the triangles A130534 (m=1), A028421 (m=2), A163932 (m=3) and A163934 (m=4). The o.g.f.s. of the right hand columns of these triangles in their turn lead to the triangles A163936 (m=1), A163937 (m=2), A163938 (m=3) and A163939 (m=4). The row sums of these four triangles lead to A001147, A001147 (minus a(0)), A001879 and A000457 which are the first four right hand columns of A001498. We checked this phenomenon for a few more values of m and found that this pattern persists: m = 5 leads to A001880, m=6 to A001881, m=7 to A038121 and m=8 to A130563 which are the next four right hand columns of A001498. So one by one all columns of the triangle of coefficients of Bessel polynomials appear. - Johannes W. Meijer, Oct 07 2009
a(n,k) also appear as coefficients of (n+1)st degree of the differential operator D:=1/t d/dt, namely D^{n+1}= Sum_{k=0..n} a(n,k) (-1)^{n-k} t^{1-(n+k)} (d^{n+1-k}/dt^{n+1-k}. - Leonid Bedratyuk, Aug 06 2010
a(n-1,k) are the coefficients when expanding (xI)^n in terms of powers of I. Let I(f)(x) := Integral_{a..x} f(t) dt, and (xI)^n := x Integral_{a..x} [ x_{n-1} Integral_{a..x_{n-1}} [ x_{n-2} Integral_{a..x_{n-2}} ... [ x_1 Integral_{a..x_1} f(t) dt ] dx_1 ] .. dx_{n-2} ] dx_{n-1}. Then: (xI)^n = Sum_{k=0..n-1} (-1)^k * a(n-1,k) * x^(n-k) * I^(n+k)(f)(x) where I^(n) denotes iterated integration. - Abdelhay Benmoussa, Apr 11 2025

			The triangle a(n, k), n >= 0, k = 0..n, begins:
  1
  1  1
  1  3   3
  1  6  15    15
  1 10  45   105    105
  1 15 105   420    945    945
  1 21 210  1260   4725  10395   10395
  1 28 378  3150  17325  62370  135135   135135
  1 36 630  6930  51975 270270  945945  2027025  2027025
  1 45 990 13860 135135 945945 4729725 16216200 34459425 34459425
  ...
And the first few Bessel polynomials are:
  y_0(x) = 1,
  y_1(x) = x + 1,
  y_2(x) = 3*x^2 + 3*x + 1,
  y_3(x) = 15*x^3 + 15*x^2 + 6*x + 1,
  y_4(x) = 105*x^4 + 105*x^3 + 45*x^2 + 10*x + 1,
  y_5(x) = 945*x^5 + 945*x^4 + 420*x^3 + 105*x^2 + 15*x + 1,
  ...
Tree counting: a(2,1)=3 for the unordered forest of m=2 plane increasing trees with n=3 vertices, namely one tree with one vertex (root) and another tree with two vertices (a root and a leaf), labeled increasingly as (1, 23), (2,13) and (3,12). - _Wolfdieter Lang_, Sep 14 2007

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

T. D. Noe, Rows n=0..50 of triangle, flattened
Alexander Alldridge, Joachim Hilgert, and Martin R. Zirnbauer, Chevalley's restriction theorem for reductive symmetric superpairs, arXiv:0812.3530 [math.RT], 2008-2009; J. Alg. 323 (4) (2010) 1159-1185 doi:10.1016/j.jalgebra.2009.11.014, Remark 3.17.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
Juan Antonio Barcelo and Anthony Carbery, On the magnitudes of compact sets in Euclidean spaces, arXiv preprint arXiv:1507.02502 [math.MG], 2015.
François Bergeron, Philippe Flajolet, and Bruno Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1922, pp. 24-48.
Alexander W. Boldyreff, Decomposition of Rational Fractions into Partial Fractions, Nat. Math. Mag. 17 (6) (1943), 261-267; coefficients (m)N(r).
Alexander Burstein and Toufik Mansour, Words restricted by patterns with at most 2 distinct letters, arXiv:math/0110056 [math.CO], 2001.
Roudy El Haddad, Repeated Integration and Explicit Formula for the n-th Integral of x^m*(ln x)^m', arXiv:2102.11723 [math.GM], 2021.
Andrew Francis and Michael Hendriksen, Counting spinal phylogenetic networks, arXiv:2502.14223 [q-bio.PE], 2025. See p. 9.
Emil Grosswald, Bessel Polynomials: Recurrence Relations, Lecture Notes Math. vol. 698, 1978, p. 18.
Cameron Jakub and Mihai Nica, Depth Degeneracy in Neural Networks: Vanishing Angles in Fully Connected ReLU Networks on Initialization, arXiv:2302.09712 [stat.ML], 2023.
Taekyun Kim, and Dae San Kim, Identities involving Bessel polynomials arising from linear differential equations, arXiv:1602.04106 [math.NT], 2016.
H. L. Krall and Orrin Frink, A new class of orthogonal polynomials, Trans. Amer. Math. Soc. 65, 100-115, 1949.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First ten rows.
B. Leclerc, Powers of staircase Schur functions and symmetric analogues of Bessel polynomials, Discrete Math., 153 (1996), 213-227.
Shi-Mei Ma, Toufik Mansour, and Matthias Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169 [math.CO], 2013.
Shi-Mei Ma, Toufik Mansour, Jean Yeh, and Yeong-Nan Yeh, Normal ordered grammars, arXiv:2404.15119 [math.CO], 2024. See p. 11.
Guillermo Navas-Palencia, On the computation of the cumulative distribution function of the Normal Inverse Gaussian distribution, arXiv:2502.16015 [math.NA], 2025. See p. 25.
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
John Riordan, Notes to N. J. A. Sloane, Jul. 1968
Florian Stober, Average case considerations for MergeInsertion, Master's Thesis, University of Stuttgart, Institute of Formal Methods in Computer Science, 2018.
Florian Stober and Armin Weiß, On the Average Case of MergeInsertion, arXiv:1905.09656 [cs.DS], 2019.
Laszlo A. Székely, Pál L. Erdős, and M. A. Steel, The combinatorics of evolutionary trees, Séminaire Lotharingien de Combinatoire, B28e (1992), 15 pp.
Juan G. Triana, Bessel polynomials by context-free grammars (Polinomios de Bessel mediante gramáticas independientes del contexto), Bistua, Univ. de Pamplona (Colombia, 2024) Vol 22, No. 2. See p. 3.
Jonas Wahl, Traces on diagram algebras II: Centralizer algebras of easy groups and new variations of the Young graph, arXiv:2009.08181 [math.RT], 2020.
Eric Weisstein's World of Mathematics, Modified Spherical Bessel Function of the Second Kind
Index entries for sequences related to Bessel functions or polynomials

Cf. A001497 (same triangle but rows read in reverse order). Other versions of this same triangle are given in A144331, A144299, A111924 and A100861.
Columns from left edge include A000217, A050534.
Columns 1-6 from right edge are A001147, A001879, A000457, A001880, A001881, A038121.
Bessel polynomials evaluated at certain x are A001515 (x=1, row sums), A000806 (x=-1), A001517 (x=2), A002119 (x=-2), A001518 (x=3), A065923 (x=-3), A065919 (x=4). Cf. A043301, A003215.
Cf. A245066 (central terms). A113025 (y_n(2*x)).

a001498 n k = a001498_tabl !! n !! k
a001498_row n = a001498_tabl !! n
a001498_tabl = map reverse a001497_tabl
-- Reinhard Zumkeller, Jul 11 2014

/* As triangle: */ [[Factorial(n+k)/(2^k*Factorial(n-k)*Factorial(k)): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 15 2016

Bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end; # explicit Bessel polynomials
Bessel := proc(n) option remember; if n <=1 then (1+x)^n else (2*n-1)*x*Bessel(n-1)+Bessel(n-2); fi; end; # recurrence for Bessel polynomials
bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end;
f := proc(n) option remember; if n <=1 then (1+x)^n else (2*n-1)*x*f(n-1)+f(n-2); fi; end;
# Alternative:
T := (n,k) -> pochhammer(n+1,k)*binomial(n,k)/2^k:
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, May 11 2018
T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)
else (n - k + 1)* T(n, k - 1) + T(n - 1, k) fi fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Oct 02 2023

max=50; Flatten[Table[(n+k)!/(2^k*(n-k)!*k!), {n, 0, Sqrt[2 max]//Ceiling}, {k, 0, n}]][[1 ;; max]] (* Jean-François Alcover, Mar 20 2011 *)

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

A001497_ser(N,t='t) = {
  my(x='x+O('x^(N+2)));
  serlaplace(deriv(exp((1-sqrt(1-2*t*x))/t),'x));
};
concat(apply(Vecrev, Vec(A001497_ser(9)))) \\ Gheorghe Coserea, Dec 27 2017

a(n, k) = (n+k)!/(2^k*(n-k)!*k!) (see Grosswald and Riordan). - Ralf Stephan, Apr 20 2004
a(n, 0)=1; a(0, k)=0, k > 0; a(n, k) = a(n-1, k) + (n-k+1) * a(n, k-1) = a(n-1, k) + (n+k-1) * a(n-1, k-1). - Len Smiley
a(n, m) = A001497(n, n-m) = A001147(m)*binomial(n+m, 2*m) for n >= m >= 0, otherwise 0.
G.f. for m-th column: (A001147(m)*x^m)/(1-x)^(2*m+1), m >= 0, where A001147(m) = double factorials (from explicit a(n, m) form).
Row polynomials y_n(x) are given by D^(n+1)(exp(t)) evaluated at t = 0, where D is the operator 1/(1-t*x)*d/dt. - Peter Bala, Nov 25 2011
G.f.: conjecture: T(0)/(1-x), where T(k) = 1 - x*y*(k+1)/(x*y*(k+1) - (1-x)^2/T(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 13 2013
Recurrence from Grosswald, p. 18, eq. (5), for the row polynomials: y_n(x) = (2*n-1)*x*y_{n-1} + y_{n-2}(x), y_{-1}(x) = 1 = y_{0} = 1, n >= 1. This becomes, for n >= 0, k = 0..n: a(n, k) = 0 for n < k (zeros not shown in the triangle), a(n, -1) = 0, a(0, 0) = 1 = a(1, 0) and otherwise a(n, k) = (2*n-1)*a(n-1, k-1) + a(n-2, k). Compare with the above given recurrences. - Wolfdieter Lang, May 11 2018
T(n, k) = Pochhammer(n+1,k)*binomial(n,k)/2^k = A113025(n,k)/2^k. - Peter Luschny, May 11 2018
a(n, k) = Sum_{i=0..min(n-1, k)} (n-i)(k-i) * a(n-1, i) where x(n) = x*(x-1)*...*(x-n+1) is the falling factorial, this equality follows directly from the operational formula we wrote in Apr 11 2025.- Abdelhay Benmoussa, May 18 2025

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

Original entry on oeis.org

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

A000806 Bessel polynomial y_n(-1).

Original entry on oeis.org

1, 0, 1, -5, 36, -329, 3655, -47844, 721315, -12310199, 234615096, -4939227215, 113836841041, -2850860253240, 77087063678521, -2238375706930349, 69466733978519340, -2294640596998068569, 80381887628910919255, -2976424482866702081004, 116160936719430292078411

Offset: 0

N. J. A. Sloane

a(n) can be seen as a subset of the unordered pairings of the first 2n integers (A001147) with forbidden pairs (i,i+1) for all i in [1,2n-1] (all adjacent integers). The circular version of this constraint is A003436. - Olivier Gérard, Feb 08 2011
|a(n)| is the number of perfect matchings in the complement of P_{2n} where P_{2n} is the path graph on 2n vertices. - Andrew Howroyd, Mar 15 2016
The unsigned version of these numbers now has its own entry: see A278990. - N. J. A. Sloane, Dec 07 2016

			For n=3, the a(3) = 5 solutions are (14) (25) (36), (14) (26) (35), (15) (24) (36), (16) (24) (35), (13) (25) (46) excluding 10 other possible pairings.
G.f. = 1 + x^2 - 5*x^3 + 36*x^4 - 329*x^5 + 3655*x^6 - 47844*x^7 + ...

G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..404 (first 101 terms from T. D. Noe)
Ron M. Adin, Arkady Berenstein, Jacob Greenstein, Jian-Rong Li, Avichai Marmor, and Yuval Roichman, Transitive and Gallai colorings, arXiv:2309.11203 [math.CO], 2023. See p. 6.
G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74. (Annotated scanned copy)
R. J. Mathar, A class of multinomial permutations avoiding object clusters, vixra:1511.0015 (2015), sequence M_{c,2}/c!.
J. Riordan, Letter to N. J. A. Sloane, Aug. 1970
Everett Sullivan, Linear chord diagrams with long chords, arXiv preprint arXiv:1611.02771 [math.CO], 2016.
J. Touchard, Nombres exponentiels et nombres de Bernoulli, Canad. J. Math., 8 (1956), 305-320.
Donovan Young, Linear k-Chord Diagrams, arXiv:2004.06921 [math.CO], 2020.
Index entries for sequences related to Bessel functions or polynomials

Polynomial coefficients are in A001498. Cf. A003436.

Cf. A001515, A101682, A278990.

I:=[0,1]; [1] cat [n le 2 select I[n] else (1-2*n)*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Apr 19 2015

A000806 := proc(n) option remember; if n<=1 then 1-n else (1-2*n)*procname(n-1)+procname(n-2); fi; end proc;
a := n -> hypergeom([n+1,-n],[],1/2): seq(simplify(a(n)),n=0..20); # Peter Luschny, Nov 10 2016

a[n_] := a[n] = (-2n+1)*a[n-1] + a[n-2]; a[0] = 1; a[1] = 0; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Nov 29 2011, after T. D. Noe *)
Table[Sum[Binomial[n, i]*(2*n-i)!/2^(n-i)*(-1)^(n-i)/n!, {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2013 *)
a[ n_] := With[ {m = If[ n<0, -n-1, n]}, (-1)^m (2 m - 1)!! Hypergeometric1F1[ -m, -2 m, -2] ]; (* Michael Somos, Jan 27 2014 *)
a[ n_] := With[ {m = If[ n<0, -n-1, n]}, Sum[ (-1)^(m - i) (2 m - i)! / (2^(m - i) i! (m - i)!), {i, 0, m}] ]; (* Michael Somos, Jan 27 2014 *)
a[ n_] := With[ {m = If[ n<0, -n-1, n]}, If[ m<1, 1, (-1)^m Numerator @ FromContinuedFraction[ Table[ (-1)^Quotient[k, 2] If[ OddQ[k], k, 1], {k, 2 m}] ] ] ]; (* Michael Somos, Jan 27 2014 *)
Table[(-1)^n (2 n - 1)!! Hypergeometric1F1[-n, -2 n, -2], {n, 0, 20}] (* Eric W. Weisstein, Nov 14 2018 *)

{a(n) = if( n<0, n = -n-1); sum(k=0, n, (2*n-k)! / (k! * (n-k)!) * (-1/2)^(n-k) )}; /* Michael Somos, Apr 02 2007 */

{a(n) = local(A); if( n<0, n = -n-1); A = sqrt(1 + 2*x + x * O(x^n)); n! * polcoeff( exp(A-1) / A, n)}; /* Michael Somos, Apr 02 2007 */

{a(n) = local(A); if( n<0, n = -n-1); n+=2; -(-1)^n * n! * polcoeff( serreverse( sum(k=1, n, k^(k-2)* x^k / k!, x * O(x^n))), n)}; /* Michael Somos, Apr 02 2007 */

{a(n) = if( n<0, n=-n-1); contfracpnqn( vector( 2*n, k, (-1)^(k\2) * if( k%2, k, 1))) [1,1] }; /* Michael Somos, Jan 27 2014 */

E.g.f.: exp(sqrt(1 + 2*x) - 1) / sqrt(1 + 2*x). - Michael Somos, Feb 16 2002
D-finite with recurrence a(n) = (-2*n+1)*a(n-1) + a(n-2). - T. D. Noe, Oct 26 2006
If y = x + Sum_{k>1} A000272(k) * x^k/k!, then y = x + Sum{k>1} a(k-2) * (-y)^k/k!. - Michael Somos, Sep 07 2005
a(-1-n) = a(n). - Michael Somos, Apr 02 2007
a(n) = Sum_{m=0..n} A001498(n,m)*(-1)^m, n>=0 (alternating row sums of Bessel triangle).
E.g.f. for unsigned version: -exp(sqrt(1-2*x)-1). - Karol A. Penson, Mar 20 2010 [gives -1, 1, 0, 1, 5, 36, 329, ... ]
E.g.f. for unsigned version: 1/(sqrt(1-2*x))*exp(sqrt(1-2*x)-1). - Sergei N. Gladkovskii, Jul 03 2012
G.f.: 1/G(0) where G(k) = 1 - x + x*(2*k+1)/(1 - x + 2*x*(k+1)/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Jul 10 2012
G.f.: 1+x/U(0)  where U(k) = 1 - x + x*(k+1)/U(k+1) ; (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Oct 06 2012
a(n) = BesselK[n+1/2,-1]/BesselK[5/2,-1]. - Vaclav Kotesovec, Aug 07 2013
|a(n)| ~ 2^(n+1/2)*n^n/exp(n+1). - Vaclav Kotesovec, Aug 07 2013
0 = a(n) * (a(n+2)) + a(n+1) * (-a(n+1) + 2*a(n+2) + a(n+3)) + a(n+2) * (-a(n+2)) for all n in Z. - Michael Somos, Jan 27 2014
a(n) = -i*(BesselK[3/2,1]*BesselI[n+3/2,-1] - BesselI[3/2,-1]*BesselK[n+3/2,1]), n>=0 for unsigned version - G. C. Greubel , Apr 19 2015
a(n) = hypergeom( [n+1, -n], [], 1/2). - Peter Luschny, Nov 10 2016
From G. C. Greubel, Aug 16 2017: (Start)
a(n) = (1/2)_{n} * (-2)^n * hypergeometric1f1(-n; -2*n; -2).
G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; -2*t/(1-t)^2). (End)

A001517 Bessel polynomials y_n(x) (see A001498) evaluated at 2.

Original entry on oeis.org

1, 3, 19, 193, 2721, 49171, 1084483, 28245729, 848456353, 28875761731, 1098127402131, 46150226651233, 2124008553358849, 106246577894593683, 5739439214861417731, 332993721039856822081, 20651350143685984386753

Offset: 0

N. J. A. Sloane

Numerators of successive convergents to e using continued fraction 1 + 2/(1 + 1/(6 + 1/(10 + 1/(14 + 1/(18 + 1/(22 + 1/26 + ...)))))).

Number of ways to use the elements of {1,...,k}, n <= k <= 2n, once each to form a collection of n lists, each having length 1 or 2. - Bob Proctor, Apr 18 2005, Jun 26 2006

L. Euler, 1737.
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 6th ed., Section 0.126, p. 2.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 0..334 (first 101 terms from T. D. Noe)
P. Bala, A note on the Catalan transform of a sequence.
J. W. L. Glaisher, On Lambert's proof of the irrationality of Pi and on the irrationality of certain other quantities, Reports of British Assoc. Adv. Sci., 1871, pp. 16-18.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 131.
D. H. Lehmer, Arithmetical periodicities of Bessel functions, Annals of Mathematics, 33 (1932): 143-150. The sequence is on page 149.
D. H. Lehmer, Review of various tables by P. Pederson, Math. Comp., 2 (1946), 68-69.
W. Mlotkowski, A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
J. Riordan, Letter to N. J. A. Sloane, Jul. 1968.
J. Riordan, Letter, Jul 06 1978.
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970.
Index entries for related partition-counting sequences
Index entries for sequences related to Bessel functions or polynomials

Essentially the same as A080893.
a(n) = A099022(n)/n!.
Partial sums: A105747.
Replace "lists" with "sets" in comment: A001515.
Cf. A001515, A001518, A002119, A053556, A053557, A243593.

A:= gfun:-rectoproc({a(n) = (4*n-2)*a(n-1) + a(n-2),a(0)=1,a(1)=3},a(n),remember):
map(A, [$0..20]); # Robert Israel, Jul 22 2015
f:=proc(n) option remember; if n = 0 then 1 elif n=1 then 3 else f(n-2)+(4*n-2)*f(n-1); fi; end;
[seq(f(n), n=0..20)]; # N. J. A. Sloane, May 09 2016
seq(simplify(KummerU(-n, -2*n, 1)), n = 0..16); # Peter Luschny, May 10 2022

Table[(2k)! Hypergeometric1F1[-k, -2k, 1]/k!, {k, 0, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)

PARI
```
a(n)=sum(k=0,n,(n+k)!/k!/(n-k)!)
```

A001517 = lambda n: hypergeometric([-n, n+1], [], -1)
[simplify(A001517(n)) for n in (0..16)] # Peter Luschny, Oct 17 2014

a(n) = Sum_{k=0..n} (n+k)!/(k!*(n-k)!) = (e/Pi)^(1/2) K_{n+1/2}(1/2).
D-finite with recurrence a(n) = (4*n-2)*a(n-1) + a(n-2), n >= 2.
a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n+k)*binomial(n,k)*A000522(n+k). - Vladeta Jovovic, Sep 30 2006
E.g.f. (for offset 1): exp(x*c(x)), where c(x)=(1-sqrt(1-4*x))/(2*x) (cf. A000108). - Vladimir Kruchinin, Aug 10 2010
G.f.: 1/Q(0), where Q(k) = 1 - x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013
a(n) = (1/n!)*Integral_{x>=0} (x*(1 + x))^n*exp(-x) dx. Expansion of exp(x) in powers of y = x*(1 - x): exp(x) = 1 + y + 3*y^2/2! + 19*y^3/3! + 193*y^4/4! + 2721*y^5/5! + .... - Peter Bala, Dec 15 2013
a(n) = exp(1/2) / sqrt(Pi) * BesselK(n+1/2, 1/2). - Vaclav Kotesovec, Mar 15 2014
a(n) ~ 2^(2*n+1/2) * n^n / exp(n-1/2). - Vaclav Kotesovec, Mar 15 2014
a(n) = hypergeom([-n, n+1], [], -1). - Peter Luschny, Oct 17 2014
From G. C. Greubel, Aug 16 2017: (Start)
a(n) = (1/2)_{n} * 4^n * hypergeometric1f1(-n; -2*n; 1).
G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; 4*t/(1-t)^2). (End)
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(n+k,k)*k!. - Ilya Gutkovskiy, Nov 24 2017
a(n) = KummerU(-n, -2*n, 1). - Peter Luschny, May 10 2022

More terms from Vladeta Jovovic, Apr 03 2000

Additional comments from Michael Somos, Jul 15 2002

A243953 E.g.f.: exp( Sum_{n>=1} A000108(n-1)x^n/n ), where A000108(n) = binomial(2n,n)/(n+1) forms the Catalan numbers.

Original entry on oeis.org

1, 1, 2, 8, 56, 592, 8512, 155584, 3456896, 90501632, 2728876544, 93143809024, 3550380249088, 149488545697792, 6890674623094784, 345131685337530368, 18664673706719019008, 1083931601731053223936, 67278418002152175960064, 4444711314548967826259968

Offset: 0

Paul D. Hanna, Jun 21 2014

			G.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 56*x^4/4! + 592*x^5/5! + 8512*x^6/6! +...
such that the logarithmic derivative of the e.g.f. equals the Catalan numbers:
log(A(x)) = x + x^2/2 + 2*x^3/3 + 5*x^4/4 + 14*x^5/5 + 42*x^6/6 + 132*x^7/7 + 429*x^8/8 +...+ A000108(n-1)*x^n/n +...
thus A'(x)/A(x) = C(x) where C(x) = 1 + x*C(x)^2.
Also, e.g.f. A(x) satisfies:
A(x) = 1 + x/A(x) + 4*(x/A(x))^2/2! + 32*(x/A(x))^3/3! + 400*(x/A(x))^4/4! + 6912*(x/A(x))^5/5! +...+ (n+1)^(n-2)*2^n*(x/A(x))^n/n! +...
If we form a table of coefficients of x^k/k! in A(x)^n, like so:
[1, 1,  2,    8,    56,    592,    8512,   155584,    3456896, ...];
[1, 2,  6,   28,   200,   2064,   28768,   511424,   11106432, ...];
[1, 3, 12,   66,   504,   5256,   72288,  1259712,   26822016, ...];
[1, 4, 20,  128,  1064,  11488,  158752,  2740480,   57517184, ...];
[1, 5, 30,  220,  2000,  22680,  319600,  5525600,  115094400, ...];
[1, 6, 42,  348,  3456,  41472,  602352, 10533024,  219321216, ...];
[1, 7, 56,  518,  5600,  71344, 1075648, 19176304,  401916032, ...];
[1, 8, 72,  736,  8624, 116736, 1835008, 33554432,  712166016, ...];
[1, 9, 90, 1008, 12744, 183168, 3009312, 56687040, 1224440064, ...]; ...
then the main diagonal equals (n+1)^(n-1) * 2^n for n>=0:
[1, 2, 12, 128, 2000, 41472, 1075648, 33554432, 1224440064, ...].
Note that Sum_{n>=0} (n+1)^(n-2) * 2^n * x^n/n! is an e.g.f. of A127670.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Cf. A000108, A127670, A243954, A213507.
Cf. A251663, A251664, A251665, A251666, A251667, A251668, A251669, A251670.
Cf. A251573, A251574, A251575, A251576, A251577, A251578, A251579, A251580.
Cf. A001515.

CoefficientList[Series[E^(1 - Sqrt[1-4*x]) * (1 + Sqrt[1-4*x])/2, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 22 2014 *)

a(n):=if n=0 then 1 else sum((n-1)!/(n-i-1)!*binomial(2*i,i)/(i+1)*a(n-i-1),i,0,n-1); /* Vladimir Kruchinin, Feb 22 2015 */

/* Explicit formula: */
{a(n)=n!*polcoeff( exp(1-sqrt(1-4*x +x*O(x^n))) * (1 + sqrt(1-4*x +x*O(x^n)))/2,n)}
for(n=0,25,print1(a(n),", "))

/* Logarithmic derivative of e.g.f. equals Catalan numbers: */
{A000108(n) = binomial(2*n,n)/(n+1)}
{a(n)=n!*polcoeff( exp(sum(m=1,n, A000108(m-1)*x^m/m)+x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))

/* From [x^n/n!] A(x)^(n+1) = (n+1)^(n-1)*2^n */
{a(n)=n!*polcoeff(x/serreverse(x*sum(m=0, n+1, (m+1)^(m-2)*(2*x)^m/m!)+x^2*O(x^n)), n)}
for(n=0,25,print1(a(n),", "))

E.g.f. A(x) satisfies:
(1) A(x) = exp(1 - sqrt(1-4*x)) * (1 + sqrt(1-4*x))/2.
(2) A(x)^2 - A(x)*A'(x) + x*A'(x)^2 = 0 (differential equation).
(3) [x^n/n!] A(x)^(n+1) = (n+1)^(n-1)*2^n for n>=0.
(4) A(x) = G(x/A(x)) such that A(x*G(x)) = G(x) = Sum_{n>=0} (n+1)^(n-2)*2^n*x^n/n!.
(5) A(x) = x / Series_Reversion(x*G(x)) where G(x) = Sum_{n>=0} (n+1)^(n-2)*2^n*x^n/n!.
(6) x = -LambertW(-2*x/A(x)) * (2 + LambertW(-2*x/A(x)))/4. [From a formula by Vaclav Kotesovec in A127670]
a(n) ~ 2^(2*n-5/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Jun 22 2014
a(n) = sum(i=0..n-1, (n-1)!/(n-i-1)!*binomial(2*i,i)/(i+1)*a(n-i-1)), a(0)=1. - Vladimir Kruchinin, Feb 22 2015
From Peter Bala, Apr 14 2017: (Start)
a(n+2) = 2^(n+1)*A001515(n).
a(n+1) = Sum_{k = 0..n} binomial(n+k-1,2*k)*2^(n-k)*(2*k)!/k!.
D-finite with recurrence a(n) = (4*n - 10)*a(n-1) + 4*a(n-2) with a(0) = a(1) = 1.
The derivative A'(x) of the e.g.f. is equal to exp(2*x*c(x)), that is, A'(x) is the Catalan transform of exp(2*x) as defined in Barry, Section 3. (End)
E.g.f. A(x) satisfies (x/A(x))' = 1/A'(x). - Alexander Burstein, Oct 31 2023

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)

A001518 Bessel polynomial y_n(3).

Original entry on oeis.org

1, 4, 37, 559, 11776, 318511, 10522639, 410701432, 18492087079, 943507142461, 53798399207356, 3390242657205889, 233980541746413697, 17551930873638233164, 1421940381306443299981, 123726365104534205331511, 11507973895102987539130504

Offset: 0

N. J. A. Sloane

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gheorghe Coserea and T. D. Noe, Table of n, a(n) for n = 0..200 (terms up to n=100 by T. D. Noe)
W. Mlotkowski, A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
Simon Plouffe, Approximations of generating functions and a few conjectures, arXiv:0911.4975 [math.NT], 2009.
J. Riordan, Letter to N. J. A. Sloane, Jul. 1968
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
Index entries for sequences related to Bessel functions or polynomials

Cf. A001515, A001517.

Polynomial coefficients are in A001498.

f:= gfun:-rectoproc({a(n)=3*(2*n-1)*a(n-1)+a(n-2),a(0)=1,a(1)=4},a(n),remember):
map(f, [$0..60]); # Robert Israel, Aug 06 2015

Table[Sum[(n+k)!*3^k/(2^k*(n-k)!*k!), {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)

x='x+O('x^33); Vec(serlaplace(exp(1/3 - 1/3 * (1-6*x)^(1/2)) / (1-6*x)^(1/2))) \\ Gheorghe Coserea, Aug 04 2015

y_n(x) = Sum_{k=0..n} (n+k)!*(x/2)^k/((n-k)!*k!).
D-finite with recurrence a(n) = 3(2n-1)*a(n-1) + a(n-2). - T. D. Noe, Oct 26 2006
G.f.: 1/Q(0), where Q(k)= 1 - x - 3*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013
a(n) = exp(1/3)*sqrt(2/(3*Pi))*BesselK(1/2+n,1/3). - Gerry Martens, Jul 22 2015
a(n) ~ sqrt(2) * 6^n * n^n / exp(n-1/3). - Vaclav Kotesovec, Jul 22 2015
E.g.f.: exp(1/3 - 1/3*(1-6*x)^(1/2)) / (1-6*x)^(1/2). (formula due to B. Salvy, see Plouffe link) - Gheorghe Coserea, Aug 06 2015
From G. C. Greubel, Aug 16 2017: (Start)
a(n) = (1/2)_{n} * 6^n * hypergeometric1f1(-n; -2*n; 2/3).
G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; 6*t/(1-t)^2). (End)

cp's OEIS Frontend

A143990 a(n) = n!*A001515(n-1) with a(0) = 1.

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A144297 BINOMIAL transform of A001515.

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A144647 Second differences of A001515 (or A144301).

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A001498 Triangle a(n,k) (n >= 0, 0 <= k <= n) of coefficients of Bessel polynomials y_n(x) (exponents in increasing order).

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

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A000806 Bessel polynomial y_n(-1).

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A243953 E.g.f.: exp( Sum_{n>=1} A000108(n-1)x^n/n ), where A000108(n) = binomial(2n,n)/(n+1) forms the Catalan numbers.