A144299 -id:A144299 - cp's OEIS frontend

A144331 Triangle b(n,k) for n >= 0, 0 <= k <= 2n, read by rows. See A144299 for definition and properties.

1, 0, 1, 1, 0, 0, 1, 3, 3, 0, 0, 0, 1, 6, 15, 15, 0, 0, 0, 0, 1, 10, 45, 105, 105, 0, 0, 0, 0, 0, 1, 15, 105, 420, 945, 945, 0, 0, 0, 0, 0, 0, 1, 21, 210, 1260, 4725, 10395, 10395, 0, 0, 0, 0, 0, 0, 0, 1, 28, 378, 3150, 17325, 62370, 135135, 135135, 0, 0, 0, 0, 0, 0

Offset: 0

David Applegate and N. J. A. Sloane, Dec 07 2008

Although this entry is the last of the versions of the underlying triangle to be added to the OEIS, for some applications it is the most important.
Row n has 2n+1 entries.
A001498 has a b-file.

			Triangle begins:
  1
  0 1 1
  0 0 1 3 3
  0 0 0 1 6 15 15
  0 0 0 0 1 10 45 105 105
  0 0 0 0 0  1 15 105 420  945  945
  0 0 0 0 0  0  1  21 210 1260 4725 10395 10395
  ...

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

Cf. A144299, A278990.
Row sums give A001515, column sums A000085.
Other versions of this triangle are given in A001497, A001498, A111924 and A100861.
See A144385 for a generalization.

a144331 n k = a144331_tabf !! n !! k
a144331_row n = a144331_tabf !! n
a144331_tabf = iterate (\xs ->
  zipWith (+) ([0] ++ xs ++ [0]) $ zipWith (*) (0:[0..]) ([0,0] ++ xs)) [1]
-- Reinhard Zumkeller, Nov 24 2014

A144331:= func< n,k | k le n-1 select 0 else Factorial(k)/(2^(k-n)*Factorial(k-n)*Factorial(2*n-k)) >;
[A144331(n,k): k in [0..2*n], n in [0..12]]; // G. C. Greubel, Oct 04 2023

Flatten[Table[PadLeft[Table[(n+k)!/(2^k*k!*(n-k)!), {k,0,n}], 2*n+1, 0], {n,0,12}]] (* Jean-François Alcover, Oct 14 2011 *)

def A144331(n, k): return 0 if kA144331(n,k) for k in range(2*n+1)] for n in range(13)]) # G. C. Greubel, Oct 04 2023

E.g.f.: Sum_{n >= 0} Sum_{k = 0..2n} b(n,k) y^n * x^k/k! = exp(x*y*(1 + x/2)).
b(n, k) = 2^(n-k)*k!/((2*n-k)!*(k-n)!).
Sum_{k=0..2*n} b(n, k) = A001515(n).
Sum_{n >= 0} b(n, k) = A000085(k).
From G. C. Greubel, Oct 04 2023: (Start)
T(n, k) = 0 for 0 <= k <= n-1, otherwise T(n, k) = k!/(2^(k-n)*(k-n)!*(2*n-k)!) for n <= k <= 2*n.
Sum_{k=0..2*n} (-1)^k * T(n, k) = A278990(n). (End)

A100861 Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 6, 3, 1, 10, 15, 1, 15, 45, 15, 1, 21, 105, 105, 1, 28, 210, 420, 105, 1, 36, 378, 1260, 945, 1, 45, 630, 3150, 4725, 945, 1, 55, 990, 6930, 17325, 10395, 1, 66, 1485, 13860, 51975, 62370, 10395, 1, 78, 2145, 25740, 135135, 270270, 135135, 1, 91, 3003, 45045, 315315, 945945, 945945, 135135

Offset: 0

Emeric Deutsch, Jan 08 2005

Row n contains 1 + floor(n/2) terms. Row sums yield A000085. T(2n,n) = T(2n-1,n-1) = (2n-1)!! (A001147).
Inverse binomial transform is triangle with T(2n,n) = (2n-1)!!, 0 otherwise. - Paul Barry, May 21 2005
Equivalently, number of involutions of n with k pairs. - Franklin T. Adams-Watters, Jun 09 2006
From Gary W. Adamson, Dec 09 2009: (Start)
If considered as an infinite lower triangular matrix (cf. A144299),
lim_{n->} A100861^n = A118930: (1, 1, 2, 4, 13, 41, ...).
(End)
Sum_{k=0..floor(n/2)} T(n,k)m^(n-2k)s^(2k) is the n-th non-central moment of the normal probability distribution with mean m and standard deviation s. - Stanislav Sykora, Jun 19 2014
Row n is the list of coefficients of the independence polynomial of the n-triangular graph. - Eric W. Weisstein, Nov 11 2016
Restating the 2nd part of the Name, row n is the list of coefficients of the matching-generating polynomial of the complete graph K_n. - Eric W. Weisstein, Apr 03 2018

			T(4, 2) = 3 because in the graph with vertex set {A, B, C, D} and edge set {AB, BC, CD, AD, AC, BD} we have the following three 2-matchings: {AB, CD},{AC, BD} and {AD, BC}.
Triangle starts:
[0] 1;
[1] 1;
[2] 1,  1;
[3] 1,  3;
[4] 1,  6,   3;
[5] 1, 10,  15;
[6] 1, 15,  45,   15;
[7] 1, 21, 105,  105;
[8] 1, 28, 210,  420, 105;
[9] 1, 36, 378, 1260, 945.
.
From _Eric W. Weisstein_, Nov 11 2016: (Start)
As polynomials:
1,
1,
1 + x,
1 + 3*x,
1 + 6*x + 3*x^2,
1 + 10*x + 15*x^2,
1 + 15*x + 45*x^2 + 15*x^3. (End)

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (1983 reprint), 10th edition, 1964, expression 22.3.11 in page 775.
C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.

T. D. Noe, Rows n = 0..100, flattened
Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
Gérard Le Caër, A new family of solvable Pearson-Dirichlet random walks, Journal of Statistical Physics 144:1 (2011), pp. 23-45.
Ji Young Choi and Jonathan D. H. Smith, On the unimodality and combinatorics of Bessel numbers, Discrete Math., 264 (2003), 45-53.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras, 2012.
John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016.
Mikael Fremling, On the modular covariance properties of composite fermions on the torus, arXiv:1810.10391 [cond-mat.str-el], 2018.
Gary R. W. Greaves, Jeven Syatriadi, and Charissa I. Utomo, Chromatic polynomials of signed graphs and dominating-vertex deletion formulae, arXiv:2407.00883 [math.CO], 2024. See p. 11.
A. Hernando, R. Hernando, A. Plastino and A. R. Plastino, The workings of the Maximum Entropy Principle in collective human behavior, arXiv preprint arXiv:1201.0905 [stat.AP], 2012.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 18.
Aaron Pollack, Exceptional theta functions, arXiv:2211.05280 [math.NT], Nov 2022. See Lemma 7.5.1.
Eric Weisstein's World of Mathematics, Complete Graph
Eric Weisstein's World of Mathematics, Independence Polynomial
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Eric Weisstein's World of Mathematics, Triangular Graph
Wikipedia, Normal distribution, section 'Moments'
Jian Zhou, Quantum deformation theory of the Airy curve and the mirror symmetry of a point, arXiv preprint arXiv:1405.5296 [math.AG], 2014.

Other versions of this same triangle are given in A144299, A001497, A001498, A111924.
Cf. A000085 (row sums).
Cf. A118930, A008619.

a100861 n k = a100861_tabf !! n !! k
a100861_row n = a100861_tabf !! n
a100861_tabf = zipWith take a008619_list a144299_tabl
-- Reinhard Zumkeller, Jan 02 2014

P[0]:=1: for n from 1 to 14 do P[n]:=sort(expand(P[n-1]+(n-1)*t*P[n-2])) od: for n from 0 to 14 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od; # yields the sequence in triangular form
# Alternative:
A100861 := proc(n,k)
    n!/k!/(n-2*k)!/2^k ;
end proc:
seq(seq(A100861(n,k),k=0..n/2),n=0..10) ; # R. J. Mathar, Aug 19 2014

Table[Table[n!/(i! 2^i (n - 2 i)!), {i, 0, Floor[n/2]}], {n, 0, 10}] // Flatten  (* Geoffrey Critzer, Mar 27 2011 *)
CoefficientList[Table[2^(n/2) (-(1/x))^(-n/2) HypergeometricU[-n/2, 1/2, -1/(2 x)], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
CoefficientList[Table[(-I)^n Sqrt[x/2]^n HermiteH[n, I/Sqrt[2 x]], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)

T(n,k)=if(k<0 || 2*k>n, 0, n!/k!/(n-2*k)!/2^k) /* Michael Somos, Jun 04 2005 */

T(n, k) = n!/(k!(n-2k)!*2^k).
E.g.f.: exp(z+tz^2/2).
G.f.: g(t, z) satisfies the differential equation g = 1 + zg + tz^2*(d/dz)(zg).
Row generating polynomial = P[n] = [-i*sqrt(t/2)]^n*H(n, i/sqrt(2t)), where H(n, x) is a Hermite polynomial and i=sqrt(-1). Row generating polynomials P[n] satisfy P[0]=1, P[n] = P[n-1] + (n-1)tP[n-2].
T(n, k) = binomial(n, 2k)(2k-1)!!. - Paul Barry, May 21 2002 [Corrected by Roland Hildebrand, Mar 06 2009]
T(n,k) = (n-2k+1)*T(n-1,k-1) + T(n-1,k). - Franklin T. Adams-Watters, Jun 09 2006
E.g.f.: 1 + (x+y*x^2/2)/(E(0)-(x+y*x^2/2)), where E(k) = 1 + (x+y*x^2/2)/(1 + (k+1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 08 2013
T(n,k) = A144299(n,k), k=0..n/2. - Reinhard Zumkeller, Jan 02 2014

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

Original entry on oeis.org

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

A099174 Triangle read by rows: coefficients of modified Hermite polynomials.

Original entry on oeis.org

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

Offset: 0

Ralf Stephan, on a suggestion of Karol A. Penson, Oct 13 2004

Absolute values of A066325.
T(n,k) is the number of involutions of {1,2,...,n}, having k fixed points (0 <= k <= n). Example: T(4,2)=6 because we have 1243,1432,1324,4231,3214 and 2134. - Emeric Deutsch, Oct 14 2006
Riordan array [exp(x^2/2),x]. - Paul Barry, Nov 06 2008
Same as triangle of Bessel numbers of second kind, B(n,k) (see Cheon et al., 2013). - N. J. A. Sloane, Sep 03 2013
The modified Hermite polynomial h(n,x) (as in the Formula section) is the numerator of the rational function given by f(n,x) = x + (n-2)/f(n-1,x), where f(x,0) = 1. - Clark Kimberling, Oct 20 2014
Second lower diagonal T(n,n-2) equals positive triangular numbers A000217 \ {0}. - M. F. Hasler, Oct 23 2014
From James East, Aug 17 2015: (Start)
T(n,k) is the number of R-classes (equivalently, L-classes) in the D-class consisting of all rank k elements of the Brauer monoid of degree n.
For n < k with n == k (mod 2), T(n,k) is the rank (minimal size of a generating set) and idempotent rank (minimal size of an idempotent generating set) of the ideal consisting of all rank <= k elements of the Brauer monoid. (End)
This array provides the coefficients of a Laplace-dual sequence H(n,x) of the Dirac delta function, delta(x), and its derivatives, formed by taking the inverse Laplace transform of these modified Hermite polynomials. H(n,x) = h(n,D) delta(x) with h(n,x) as in the examples and the lowering and raising operators L = -x and R = -x + D = -x + d/dx such that L H(n,x) = n * H(n-1,x) and R H(n,x) = H(n+1,x). The e.g.f. is exp[t H(.,x)] = e^(t^2/2) e^(t D) delta(x) = e^(t^2/2) delta(x+t). - Tom Copeland, Oct 02 2016
Antidiagonals of this entry are rows of A001497. - Tom Copeland, Oct 04 2016
This triangle is the reverse of that in Table 2 on p. 7 of the Artioli et al. paper and Table 6.2 on p. 234 of Licciardi's thesis, with associations to the telephone numbers. - Tom Copeland, Jun 18 2018 and Jul 08 2018
See A344678 for connections to a Heisenberg-Weyl algebra of differential operators, matching and independent edge sets of the regular n-simplices with partially labeled vertices, and telephone switchboard scenarios. - Tom Copeland, Jun 02 2021

			h(0,x) = 1
h(1,x) = x
h(2,x) = x^2 + 1
h(3,x) = x^3 + 3*x
h(4,x) = x^4 + 6*x^2 + 3
h(5,x) = x^5 + 10*x^3 + 15*x
h(6,x) = x^6 + 15*x^4 + 45*x^2 + 15
From _Paul Barry_, Nov 06 2008: (Start)
Triangle begins
   1,
   0,  1,
   1,  0,  1,
   0,  3,  0,  1,
   3,  0,  6,  0,  1,
   0, 15,  0, 10,  0,  1,
  15,  0, 45,  0, 15,  0,  1
Production array starts
  0, 1,
  1, 0, 1,
  0, 2, 0, 1,
  0, 0, 3, 0, 1,
  0, 0, 0, 4, 0, 1,
  0, 0, 0, 0, 5, 0, 1 (End)

Alois P. Heinz, Rows n = 0..150, flattened
M. Artioli, G. Dattoli, S. Licciardi, and S. Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017.
Paul Barry, Riordan array, orthogonal polynomials as moments, and Hankel transforms, arXiv:1102.0921 [math.CO], 2011.
G.-S. Cheon, J.-H. Jung and L. W. Shapiro, Generalized Bessel numbers and some combinatorial settings, Discrete Math., 313 (2013), 2127-2138.
T. Copeland, Juggling Zeros in the Matrix (Example II), 2020.
James East and Robert D. Gray, Diagram monoids and Graham-Houghton graphs: idempotents and generating sets of ideals, arXiv:1404.2359 [math.GR], 2014. See Theorem 8.4 and Table 7. - _James East_, Aug 17 2015
A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory, arXiv:quant-ph/0409152, 2004.
Alexander Kreinin, Combinatorial Properties of Mills' Ratio, arXiv:1405.5852 [math.CO], 2014. See Table 2. - _N. J. A. Sloane_, May 29 2014
S. Licciardi, Umbral Calculus, a Different Mathematical Language, arXiv:1803.03108 [math.CA], 2018.
R. Paris, A uniform asymptotic expansion for the incomplete gamma function, Journal of Computational and Applied Mathematics, 148 (2002), p. 223-239 (See p. 329 and A137286. From _Tom Copeland_, Jan 03 2016).
R. Sazdanovic, A categorification of the polynomial ring, slide presentation, 2011
S. Yang and Z. Qiao, The Bessel numbers and Bessel matrices, Jrn. Math. Rsch. and Exposition, July 2011, Vol. 31, No. 4, pp.627-636. DOI:10.3770/j.issn:1000-341X.2011.04.006.

Row sums (polynomial values at x=1) are A000085.
Polynomial values: A005425 (x=2), A202834 (x=3), A202879(x=4).
Cf. A000217, A001147, A059343, A060821, A066325.
Cf. A000384, A014105, A034839, A049403, A096713, A100861, A104556, A122848, A130757, A176230, A176231.
Cf. A137286.
Cf. A001497.
Cf. A111062, A344678.

T:=proc(n,k) if n-k mod 2 = 0 then n!/2^((n-k)/2)/((n-k)/2)!/k! else 0 fi end: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form; Emeric Deutsch, Oct 14 2006

nn=10;a=y x+x^2/2!;Range[0,nn]!CoefficientList[Series[Exp[a],{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, May 08 2012 *)
H[0, x_] = 1; H[1, x_] := x; H[n_, x_] := H[n, x] = x*H[n-1, x]-(n-1)* H[n-2, x]; Table[CoefficientList[H[n, x], x], {n, 0, 11}] // Flatten // Abs (* Jean-François Alcover, May 23 2016 *)
T[ n_, k_] := If[ n < 0, 0, Coefficient[HermiteH[n, x I/Sqrt[2]] (Sqrt[1/2]/I)^n, x, k]]; (* Michael Somos, May 10 2019 *)

T(n,k)=if(k<=n && k==Mod(n,2), n!/k!/(k=(n-k)/2)!>>k) \\ M. F. Hasler, Oct 23 2014

import sympy
from sympy import Poly
from sympy.abc import x, y
def H(n, x): return 1 if n==0 else x if n==1 else x*H(n - 1, x) - (n - 1)*H(n - 2, x)
def a(n): return [abs(cf) for cf in Poly(H(n, x), x).all_coeffs()[::-1]]
for n in range(21): print(a(n)) # Indranil Ghosh, May 26 2017

def Trow(n: int) -> list[int]:
    row: list[int] = [0] * (n + 1); row[n] = 1
    for k in range(n - 2, -1, -2):
        row[k] = (row[k + 2] * (k + 2) * (k + 1)) // (n - k)
    return row  # Peter Luschny, Jan 08 2023

def A099174_triangle(dim):
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(k+1)*M[n-1,k+1]
    return M
A099174_triangle(9)  # Peter Luschny, Oct 06 2012

h(k, x) = (-I/sqrt(2))^k * H(k, I*x/sqrt(2)), H(n, x) the Hermite polynomials (A060821, A059343).
T(n,k) = n!/(2^((n-k)/2)*((n-k)/2)!k!) if n-k >= 0 is even; 0 otherwise. - Emeric Deutsch, Oct 14 2006
G.f.: 1/(1-x*y-x^2/(1-x*y-2*x^2/(1-x*y-3*x^2/(1-x*y-4*x^2/(1-... (continued fraction). - Paul Barry, Apr 10 2009
E.g.f.: exp(y*x + x^2/2). - Geoffrey Critzer, May 08 2012
Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n >= 1 T(n,k) = T(n-1,k-1) + (k+1)*T(n-1,k+1). - Peter Luschny, Oct 06 2012
T(n+2,n) = A000217(n+1), n >= 0. - M. F. Hasler, Oct 23 2014
The row polynomials P(n,x) = (a. + x)^n, umbrally evaluated with (a.)^n = a_n = aerated A001147, are an Appell sequence with dP(n,x)/dx = n * P(n-1,x). The umbral compositional inverses (cf. A001147) of these polynomials are given by the same polynomials signed, A066325. - Tom Copeland, Nov 15 2014
From Tom Copeland, Dec 13 2015: (Start)
The odd rows are (2x^2)^n x n! L(n,-1/(2x^2),1/2), and the even, (2x^2)^n n! L(n,-1/(2x^2),-1/2) in sequence with n= 0,1,2,... and L(n,x,a) = Sum_{k=0..n} binomial(n+a,k+a) (-x)^k/k!, the associated Laguerre polynomial of order a. The odd rows are related to A130757, and the even to A176230 and A176231. Other versions of this entry are A122848, A049403, A096713 and A104556, and reversed A100861, A144299, A111924. With each non-vanishing diagonal divided by its initial element A001147(n), this array becomes reversed, aerated A034839.
Create four shift and stretch matrices S1,S2,S3, and S4 with all elements zero except S1(2n,n) = 1 for n >= 1, S2(n,2n) = 1 for n >= 0, S3(2n+1,n) = 1 for n >= 1, and S4(n,2n+1) = 1 for n >= 0. Then this entry's lower triangular matrix is T = Id + S1 * (A176230-Id) * S2 + S3 * (unsigned A130757-Id) * S4 with Id the identity matrix. The sandwiched matrices have infinitesimal generators with the nonvanishing subdiagonals A000384(n>0) and A014105(n>0).
As an Appell sequence, the lowering and raising operators are L = D and R = x + dlog(exp(D^2/2))/dD = x + D, where D = d/dx, L h(n,x) = n h(n-1,x), and R h(n,x) = h(n+1,x), so R^n 1 = h(n,x). The fundamental moment sequence has the e.g.f. e^(t^2/2) with coefficients a(n) = aerated A001147, i.e., h(n,x) = (a. + x)^n, as noted above. The raising operator R as a matrix acting on o.g.f.s (formal power series) is the transpose of the production matrix P below, i.e., (1,x,x^2,...)(P^T)^n (1,0,0,...)^T = h(n,x).
For characterization as a Riordan array and associations to combinatorial structures, see the Barry link and the Yang and Qiao reference. For relations to projective modules, see the Sazdanovic link.
(End)
From the Appell formalism, e^(D^2/2) x^n = h_n(x), the n-th row polynomial listed below, and e^(-D^2/2) x^n = u_n(x), the n-th row polynomial of A066325. Then R = e^(D^2/2) * x * e^(-D^2/2) is another representation of the raising operator, implied by the umbral compositional inverse relation h_n(u.(x)) = x^n. - Tom Copeland, Oct 02 2016
h_n(x) = p_n(x-1), where p_n(x) are the polynomials of A111062, related to the telephone numbers A000085. - Tom Copeland, Jun 26 2018
From Tom Copeland, Jun 06 2021: (Start)
In the power basis x^n, the matrix infinitesimal generator M = A132440^2/2, when acting on a row vector for an o.g.f., is the matrix representation for the differential operator D^2/2.
e^{M} gives the coefficients of the Hermite polynomials of this entry.
The only nonvanishing subdiagonal of M, the second subdiagonal (1,3,6,10,...), gives, aside from the initial 0, the triangular numbers A000217, the number of edges of the n-dimensional simplices with (n+1) vertices. The perfect matchings of these simplices are the aerated odd double factorials A001147 noted above, the moments for the Hermite polynomials.
The polynomials are also generated from A036040 with x[1] = x, x[2] = 1, and the other indeterminates equal to zero. (End)

A130757 Triangular table of coefficients of Laguerre-Sonin polynomials n!2^nLag(n,x/2,1/2) of order 1/2.

Original entry on oeis.org

1, 3, -1, 15, -10, 1, 105, -105, 21, -1, 945, -1260, 378, -36, 1, 10395, -17325, 6930, -990, 55, -1, 135135, -270270, 135135, -25740, 2145, -78, 1, 2027025, -4729725, 2837835, -675675, 75075, -4095, 105, -1, 34459425, -91891800, 64324260, -18378360, 2552550, -185640, 7140, -136

Offset: 0

Wolfdieter Lang, Jul 13 2007

These polynomials appear in the radial l=0 (s) wave functions of the isotropic three-dimensional harmonic quantum oscillator with the dimensionless variable x=(r/L)^2 with r>=0 and L^2=h/(m*f0). h is Planck's constant and m and f0 are the mass and the frequency of the oscillator.
From Tom Copeland, Dec 13 2015: (Start)
See A099174 for relations to the Hermite polynomials and the link in A176230 for operator relations. The infinitesimal generator for this matrix contains A014105.
The row polynomials are P(n,x) = 2^n n! Lag(n,x/2,1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q, with raising operator R = -x^(-2) [x^(3/2) (1 - 2D)]^2 = 3 - x + (4x - 6) D - 4x D^2 with D = d/dx, i.e., R P(n,x) - P(n+1,x). A matrix reresentation of R acting on an o.g.f. (formal power series) is given by the transpose of the production matrix below. The diagonal corresponds to (3 + 4 xD) x^n = (3 + 4n) x^n; the upper diagonal, to -x x^n = -x^(n+1); and the lower diagonal, to (-6 - 4 xD) D x^n = -n (6 + 4(n-1)) x^(n-1), the sequence A002943. See A176230 for a similar relation.
The triangles of Bessel numbers entries A122848, A049403, A096713, A104556 contain these polynomials as even or odd rows. Also the aerated version A099174 and A066325. Reversed, these entries are A100861, A144299, A111924.
(End)
Exponential Riordan array [1/(1-2x)^(3/2), -x/(1-2x)]. - Paul Barry, Mar 07 2017

			[1]; [3,-1]; [15,-10,1]; [105,-105,21,-1]; [945,-1260,378,-36,1]; ...

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 775, 22.3.9.
T. Copeland, Juggling Zeros in the Matrix: Example II, 2020.
Wolfdieter Lang, First ten rows and more.

Cf. A021009 (Coefficient table of n!*L(n, 0, x)).
Row sums (signed) give A131441. Row sums (unsigned) give A066224.
Cf. A001147, A002943, A014105, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299, A176230.

seq(seq(n!*2^(n-m)*(-1)^m*binomial(n+1/2,n-m)/m!,m=0..n),n=0..10); # Robert Israel, Dec 25 2015

Table[n! (2^(n - m)) ((-1)^m) Binomial[n + 1/2, n - m]/m!, {n, 0, 8}, {m, 0, n}] // Flatten (* Michael De Vlieger, Dec 24 2015 *)

a(n,m) = n!*(2^(n-m))*L(1/2,n,m) with L(1/2,n,m) = ((-1)^m)*binomial(n+1/2,n-m)/m!, n >= m >= 0, otherwise 0.
Let IP be the lower triangular matrix with its first subdiagonal equal to the first subdiagonal (cf. A014105) of this entry's unsigned matrix M and with all other elements equal to zero. Then IP is the infinitesimal generator of M, i.e., M = exp(IP). - Tom Copeland, Dec 12 2015
From Tom Copeland, Dec 14 2015: (Start)
Production matrix is
   3,   -1;
  -6,    7,   -1;
   0,  -20,   11,   -1;
   0,    0,  -42,   15,   -1;
   0,    0,    0,  -72,   19,   -1;
   0,    0,    0,    0, -110,   23,   -1;
   0,    0,    0,    0,    0, -156,   27,   -1;
   0,    0,    0,    0,    0,    0, -210,   31,   -1;
   0,    0,    0,    0,    0,    0,    0, -272,   35,   -1;
  ... (End)

Title formula corrected by Tom Copeland, Dec 12 2015

A118930 E.g.f.: A(x) = exp( Sum_{n>=0} x^(2^n)/2^(2^n-1) ).

Original entry on oeis.org

1, 1, 2, 4, 13, 41, 166, 652, 3494, 18118, 114076, 681176, 5016892, 35377564, 288204008, 2232198256, 21124254181, 191779964597, 2011347229114, 19840403629108, 231266808172181, 2553719667653281, 31743603728993542

Offset: 0

Paul D. Hanna, May 06 2006

nonn

Equals invariant column vector V that satisfies matrix product A100861*V = V, where Bessel numbers A100861(n,k) = n!/[k!(n-2k)!*2^k] give the number of k-matchings of the complete graph K(n).

Equals Lim_{n->inf.} A144299^n, if A144299 is considered an infinite lower triangular matrix. - Gary W. Adamson, Dec 08 2008

			E.g.f. A(x) = exp( x + x^2/2 + x^4/2^3 + x^8/2^7 + x^16/2^15 +...)
= 1 + 1*x + 2*x^2/2! + 4*x^3/3! + 13*x^4/4! + 41*x^5/5!+ 166*x^6/6!+...
Using coefficients A100861(n,k) = n!/[k!(n-2k)!*2^k]:
a(5) = 1*a(0) +10*a(1) +15*a(2) = 1*1 +10*1 +15*2 = 41.
a(6) = 1*a(0) +15*a(1) +45*a(2) +15*a(3) = 1*1 +15*1 +45*2 +15*4 = 166.

Cf. A100861; variants: A118932, A118935.
Equals row sums of triangle A152685. - Gary W. Adamson, Dec 10 2008
Cf. A144299. - Gary W. Adamson, Dec 08 2008

A118930 := proc(n)
    option remember;
    if n<= 1 then
        1 ;
    else
        n!*add(procname(k)/k!/(n-2*k)!/2^k,k=0..n/2) ;
    end if;
end proc;
seq(A118930(n),n=0..10) ; # R. J. Mathar, Aug 19 2014

a[n_] := a[n] = If[n==0, 1, Sum[Binomial[n, 2k] (2k-1)!! a[k], {k, 0, n/2}]];
a /@ Range[0, 22] (* Jean-François Alcover, Mar 26 2020 *)

{a(n)=if(n==0,1,sum(k=0,n\2,n!/(k!*(n-2*k)!*2^k)*a(k)))}
for(n=0,30,print1(a(n),", "))

/* Defined by E.G.F.: */
{a(n)=n!*polcoeff( exp(sum(k=0,#binary(n),x^(2^k)/2^(2^k-1))+x*O(x^n)),n,x)}
for(n=0,30,print1(a(n),", "))

a(n) = Sum_{k=0..[n/2]} n!/[k!*(n-2*k)!*2^k] * a(k), with a(0)=1. a(n) = Sum_{k=0..[n/2]} A100861(n,k)*a(k), with a(0)=1.

A144643 Triangle read by rows: T(n,k) = number of partitions of [1..k] into n nonempty clumps of sizes 1, 2, 3 or 4 (n >= 0, 0 <= k <= 4n).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 0, 1, 3, 7, 15, 25, 35, 35, 0, 0, 0, 1, 6, 25, 90, 280, 770, 1855, 3675, 5775, 5775, 0, 0, 0, 0, 1, 10, 65, 350, 1645, 6930, 26425, 90475, 275275, 725725, 1576575, 2627625, 2627625, 0, 0, 0, 0, 0, 1, 15, 140, 1050, 6825, 39795, 211750, 1033725, 4629625

Offset: 0

David Applegate and N. J. A. Sloane, Jan 25 2009

			Irregular triangle begins:
  1;
  0, 1, 1, 1, 1;
  0, 0, 1, 3, 7, 15, 25,  35,  35;
  0, 0, 0, 1, 6, 25, 90, 280, 770, 1855, 3675, 5775, 5775;
  ...

G. C. Greubel, Rows n = 0..25 of the irregular triangle, flattened
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.

Row sums give A144508.
See A144644 and A144645 for other versions.
Cf. A144299, A144385.

function t(n,k)
  if k eq n then return 1;
  elif k le n-1 or n le 0 then return 0;
  else return (&+[Binomial(k-1,j)*t(n-1,k-j-1): j in [0..3]]);
  end if;
end function;
A144643:= func< n,k | t(n,k) >;
[A144643(n,k): k in [0..4*n], n in [0..8]]; // G. C. Greubel, Oct 11 2023

T := proc(n, k) option remember;
if n = k then 1;
elif k < n then 0;
elif n < 1 then 0;
else T(n - 1, k - 1) + (k - 1)*T(n - 1, k - 2) + 1/2*(k - 1)*(k - 2)*T(n - 1, k - 3) + 1/6*(k - 1)*(k - 2)*(k - 3)*T(n - 1, k - 4);
end if;
end proc;

T[n_, k_]:= T[n, k]= Which[n==k, 1, kJean-François Alcover, Mar 20 2014, after Maple *)
Table[BellY[k, n, {1,1,1,1}], {n,0,12}, {k,0,4*n}]//Flatten (* G. C. Greubel, Oct 11 2023 *)

@CachedFunction
def t(n,k):
    if (k==n): return 1
    elif (kA144643(n,k): return t(n,k)
flatten([[A144643(n,k) for k in range(4*n+1)] for n in range(13)]) # G. C. Greubel, Oct 11 2023

T(n, k) = Sum_{j=0..3} binomial(k-1, j) * T(n-1, k-j-1), with T(n, n) = 1, T(n, k) = 0 if n < 1 or n > k.

Sum_{k=0..4*n} T(n, k) = A144508(n).

A176230 Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].

Original entry on oeis.org

1, 1, 1, 3, 6, 1, 15, 45, 15, 1, 105, 420, 210, 28, 1, 945, 4725, 3150, 630, 45, 1, 10395, 62370, 51975, 13860, 1485, 66, 1, 135135, 945945, 945945, 315315, 45045, 3003, 91, 1, 2027025, 16216200, 18918900, 7567560, 1351350, 120120, 5460, 120, 1, 34459425

Offset: 0

Paul Barry, Apr 12 2010

Row sums are A066223. Reverse of A119743. Inverse is alternating sign version.
Diagonal sums are essentially A025164.
From Tom Copeland, Dec 13 2015: (Start)
See A099174 for relations to the Hermite polynomials and the link for operator relations, including the infinitesimal generator containing A000384.
Row polynomials are 2^n n! Lag(n,-x/2,-1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q.
The triangles of Bessel numbers entries A122848, A049403, A096713, A104556 contain these polynomials as even or odd rows. Also the aerated version A099174 and A066325. Reversed, these entries are A100861, A144299, A111924.
Divided along the diagonals by the initial element (A001147) of the diagonal, this matrix becomes the even rows of A034839.
(End)
The first few rows appear in expansions related to the Dedekind eta function on pp. 537-538 of the Chan et al. link. - Tom Copeland, Dec 14 2016

			Triangle begins
        1,
        1,        1,
        3,        6,        1,
       15,       45,       15,       1,
      105,      420,      210,      28,       1,
      945,     4725,     3150,     630,      45,      1,
    10395,    62370,    51975,   13860,    1485,     66,    1,
   135135,   945945,   945945,  315315,   45045,   3003,   91,   1,
  2027025, 16216200, 18918900, 7567560, 1351350, 120120, 5460, 120, 1
Production matrix is
  1,  1,
  2,  5,  1,
  0, 12,  9,  1,
  0,  0, 30, 13,  1,
  0,  0,  0, 56, 17,   1,
  0,  0,  0,  0, 90,  21,   1,
  0,  0,  0,  0,  0, 132,  25,   1,
  0,  0,  0,  0,  0,   0, 182,  29,  1,
  0,  0,  0,  0,  0,   0,   0, 240, 33, 1.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Bala, Generalized Dobinski formulas
H. Chan, S. Cooper, and P. Toh, The 26th power of Dedekind's eta function Advances in Mathematics, 207 (2006) 532-543.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras, 2012.
Tom Copeland, Juggling Zeros in the Matrix (Example II), 2020.

Cf. A113278.

Cf. A000384, A001147, A034839, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299.

ser := n -> series(KummerU(-n, 1/2, x), x, n+1):
seq(seq((-2)^(n-k)*coeff(ser(n), x, k), k=0..n), n=0..8); # Peter Luschny, Jan 18 2020

t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); u[n_, k_] := t[2 n, k + n]; Table[ u[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jan 14 2011 *)

Number triangle T(n,k) = (2n)!/((2k)!(n-k)!2^(n-k)).
T(n,k) = A122848(2n,k+n). - R. J. Mathar, Jan 14 2011
[x^(1/2)(1+2D)]^2 p(n,x)= p(n+1,x) and [D/(1+2D)]p(n,x)= n p(n-1,x) for the row polynomials of T, with D=d/dx. - Tom Copeland, Dec 26 2012
E.g.f.: exp[t*x/(1-2x)]/(1-2x)^(1/2). - Tom Copeland , Dec 10 2013
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} (2*k+1)*(2*k+3)*...*(2*k+1+2*(n-1))*(x/2)^k/k!. Cf. A113278. - Peter Bala, Jun 23 2014
The raising operator in my 2012 formula expanded is R = [x^(1/2)(1+2D)]^2 = 1 + x + (2 + 4x) D + 4x D^2, which in matrix form acting on an o.g.f. (formal power series) is the transpose of the production array below. The linear term x is the diagonal of ones after transposition. The main diagonal comes from (1 + 4xD) x^n = (1 + 4n) x^n. The last diagonal comes from (2 D + 4 x D^2) x^n = (2 + 4 xD) D x^n = n * (2 + 4(n-1)) x^(n-1). - Tom Copeland, Dec 13 2015
T(n, k) = (-2)^(n-k)*[x^k] KummerU(-n, 1/2, x). - Peter Luschny, Jan 18 2020

A359760 Triangle read by rows. The Kummer triangle, the coefficients of the Kummer polynomials. K(n, k) = binomial(n, k) * oddfactorial(k/2) if k is even, otherwise 0, where oddfactorial(z) := (2z)!/(2^zz!).

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Jan 13 2023

The Kummer numbers K(n, k) are a refinement of the oddfactorial numbers (A001147) in the sense that they are the coefficients of polynomials K(n, x) = Sum_{n..k} K(n, k) * x^k that take the value oddfactorial(n) at x = 1. The coefficients of x^n are the aerated oddfactorial numbers A123023.

These numbers appear in many different versions (see the crossrefs). They are the coefficients of the Chebyshev-Hermite polynomials in signed form when ordered in decreasing powers. Our exposition is based on the seminal paper by Kummer, which preceded the work of Chebyshev and Hermite for more than 20 years. They are also referred to as Bessel numbers of the second kind (Mansour et al.) when the odd powers are omitted.

			Triangle K(n, k) starts:
 [0] 1;
 [1] 1, 0;
 [2] 1, 0,  1;
 [3] 1, 0,  3, 0;
 [4] 1, 0,  6, 0,   3;
 [5] 1, 0, 10, 0,  15, 0;
 [6] 1, 0, 15, 0,  45, 0,   15;
 [7] 1, 0, 21, 0, 105, 0,  105, 0;
 [8] 1, 0, 28, 0, 210, 0,  420, 0, 105;
 [9] 1, 0, 36, 0, 378, 0, 1260, 0, 945, 0;

John Riordan, Introduction to Combinatorial Analysis, Dover (2002), pp. 85-86.

Pierre Humbert, Monographie des polynômes de Kummer, Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Serie 5, Volume 1 (1922), pp. 81-92.
E. E. Kummer, Über die hypergeometrische Reihe, Journal für die reine und angewandte Mathematik 15 (1836): 39-83.
T. Mansour, M. Schork and M. Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
Ladislav Truksa, Hypergeometric orthogonal systems of polynomials III, Aktuárské vědy, Vol. 2 (1931), No. 4, 177-203, (see p.200).

Variants: Signed version: A073278. Other variants are the irregular triangle A100861 with zeros deleted, A066325 and A099174 with reversed rows, A111924, A144299, A104556.
Cf. A000085, A001147, A123023, A047974, A115329, A002522, A056107, A359739,
A001879, A001464, A005425, A344501, A123022, A359761.

oddfactorial := proc(z) (2*z)! / (2^z*z!) end:
K := (n, k) -> ifelse(irem(k, 2) = 1, 0, binomial(n, k) * oddfactorial(k/2)):
seq(seq(K(n, k), k = 0..n), n = 0..11);
# Alternative, as coefficients of polynomials:
p := (n, x) -> 2^(n/2)*(-1/x^2)^(-n/2)*KummerU(-n/2, 1/2, -1/(2*x^2)):
seq(print(seq(coeff(simplify(p(n, x)), x, k), k = 0..n)), n = 0 ..9);
# Using the exponential generating function:
egf := exp(x + (t*x)^2 / 2): ser := series(egf, x, 12):
seq(print(seq(coeff(n! * coeff(ser, x, n), t, k), k = 0..n)), n = 0..9);

K[n_, k_] := K[n, k] = Which[OddQ[k], 0, k == 0, 1, n == k, K[n - 1, n - 2], True, K[n - 1, k] n/(n - k)];
Table[K[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 25 2023 *)

from functools import cache
@cache
def K(n: int, k: int) -> int:
    if k %  2: return 0
    if n <  3: return 1
    if n == k: return K(n - 1, n - 2)
    return (K(n - 1, k) * n) // (n - k)
for n in range(10): print([K(n, k) for k in range(n + 1)])

Let p(n, x) = 2^(n/2)*(-1/x^2)^(-n/2)*KummerU(-n/2, 1/2, -1/(2*x^2)).
p(n, 1) = A000085(n); p(n, sqrt(2)) = A047974(n); p(n, 2) = A115329(n);
p(2, n) = A002522(n) (n >= 1); p(3, n) = A056107(n) (n >= 1);
p(n, n) = A359739(n) (n >= 1); 2^n*p(n, 1/2) = A005425(n).
K(n, k) = [x^k] p(n, x).
K(n, k) = [t^k] (n! * [x^n] exp(x + (t*x)^2 / 2)).
K(n, n) = A123023(n).
K(n, n-1) = A123023(n + 1).
K(2*n, 2*n) = A001147(n).
K(4*n, 2*n) = A359761, the central terms without zeros.
K(2*n+2, 2*n) = A001879.
Sum_{k=0..n} (-1)^n * i^k * K(n, k) = A001464(n), ((the number of even involutions) - (the number of odd involutions) in the symmetric group S_n (Robert Israel)).
Sum_{k=0..n} Sum_{j=0..k} K(n, j) = A000085(n + 1).
For a recursion see the Python program.

cp's OEIS Frontend

A144331 Triangle b(n,k) for n >= 0, 0 <= k <= 2n, read by rows. See A144299 for definition and properties.

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A100861 Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).

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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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A099174 Triangle read by rows: coefficients of modified Hermite polynomials.

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A130757 Triangular table of coefficients of Laguerre-Sonin polynomials n!2^nLag(n,x/2,1/2) of order 1/2.

A359760 Triangle read by rows. The Kummer triangle, the coefficients of the Kummer polynomials. K(n, k) = binomial(n, k) * oddfactorial(k/2) if k is even, otherwise 0, where oddfactorial(z) := (2z)!/(2^zz!).