%I #157 Aug 28 2024 12:41:10

%S 1,1,1,1,3,3,1,6,15,15,1,10,45,105,105,1,15,105,420,945,945,1,21,210,

%T 1260,4725,10395,10395,1,28,378,3150,17325,62370,135135,135135,1,36,

%U 630,6930,51975,270270,945945,2027025,2027025,1,45,990,13860,135135,945945,4729725,16216200,34459425,34459425

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

%C 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).

%C Also called Bessel numbers of first kind.

%C 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

%C 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

%C Related to Hermite polynomials by a(n,k)=(-1)^k*A060821(n+k, n-k)/2^n. - _Paul Barry_, Aug 28 2005

%C 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.

%C 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

%C 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

%C 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

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

%H T. D. Noe, <a href="/A001498/b001498.txt">Rows n=0..50 of triangle, flattened</a>

%H A. Alldrige, J. Hilgert, and M. R. Zirnbauer, <a href="https://arxiv.org/abs/0812.3530">Chevalley's restriction theorem for reductive symmetric superpairs</a>, 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.

%H David Applegate and N. J. A. Sloane, <a href="http://arxiv.org/abs/0907.0513">The Gift Exchange Problem</a>, arXiv:0907.0513 [math.CO], 2009.

%H Juan Antonio Barcelo and Anthony Carbery, <a href="http://arxiv.org/abs/1507.02502">On the magnitudes of compact sets in Euclidean spaces</a>, arXiv preprint arXiv:1507.02502 [math.MG], 2015.

%H F. Bergeron, Ph. Flajolet and B. Salvy, <a href="http://algo.inria.fr/flajolet/Publications/BeFlSa92.pdf">Varieties of Increasing Trees</a>, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1922, pp. 24-48.

%H Alexander W. Boldyreff, <a href="http://www.jstor.org/stable/3028007">Decomposition of Rational Fractions into Partial Fractions</a>, Nat. Math. Mag. 17 (6) (1943), 261-267; coefficients (m)N(r).

%H E. Grosswald, <a href="http://dx.doi.org/10.1007/BFb0063138">Bessel Polynomials: Recurrence Relations</a>, Lecture Notes Math. vol. 698, 1978, p. 18.

%H Alexander Burstein and Toufik Mansour, <a href="http://arXiv.org/abs/math.CO/0110056">Words restricted by patterns with at most 2 distinct letters</a>, arXiv:math/0110056 [math.CO], 2001.

%H Cameron Jakub and Mihai Nica, <a href="https://arxiv.org/abs/2302.09712">Depth Degeneracy in Neural Networks: Vanishing Angles in Fully Connected ReLU Networks on Initialization</a>, arXiv:2302.09712 [stat.ML], 2023.

%H Taekyun Kim, and Dae San Kim, <a href="http://arxiv.org/abs/1602.04106">Identities involving Bessel polynomials arising from linear differential equations</a>, arXiv:1602.04106 [math.NT], 2016.

%H H. L. Krall and Orrin Frink, <a href="http://dx.doi.org/10.1090/S0002-9947-1949-0028473-1">A new class of orthogonal polynomials</a>, Trans. Amer. Math. Soc. 65, 100-115, 1949.

%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

%H Wolfdieter Lang, <a href="/A001498/a001498.txt">First ten rows.</a>

%H B. Leclerc, <a href="http://dx.doi.org/10.1016/0012-365X(95)00138-M">Powers of staircase Schur functions and symmetric analogues of Bessel polynomials</a>, Discrete Math., 153 (1996), 213-227.

%H Shi-Mei Ma, Toufik Mansour, and Matthias Schork. <a href="http://arxiv.org/abs/1308.0169">Normal ordering problem and the extensions of the Stirling grammar</a>, arXiv preprint arXiv:1308.0169 [math.CO], 2013.

%H Shi-Mei Ma, Toufik Mansour, Jean Yeh, and Yeong-Nan Yeh, <a href="https://arxiv.org/abs/2404.15119">Normal ordered grammars</a>, arXiv:2404.15119 [math.CO], 2024. See p. 11.

%H Andrew Elvey Price and Alan D. Sokal, <a href="https://arxiv.org/abs/2001.01468">Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials</a>, arXiv:2001.01468 [math.CO], 2020.

%H John Riordan, <a href="/A001820/a001820.pdf">Notes to N. J. A. Sloane, Jul. 1968</a>

%H Florian Stober, <a href="https://elib.uni-stuttgart.de/bitstream/11682/10261/1/Bachelorarbeit_MergeInsertion_2018-11-28.pdf">Average case considerations for MergeInsertion</a>, Master's Thesis, University of Stuttgart, Institute of Formal Methods in Computer Science, 2018.

%H Florian Stober and Armin Weiß, <a href="https://arxiv.org/abs/1905.09656">On the Average Case of MergeInsertion</a>, arXiv:1905.09656 [cs.DS], 2019.

%H Laszlo A. Székely, Pál L. Erdős, and M. A. Steel, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28szekely.html">The combinatorics of evolutionary trees</a>, Séminaire Lotharingien de Combinatoire, B28e (1992), 15 pp.

%H Juan G. Triana, <a href="https://ojs.unipamplona.edu.co/index.php/bistua/article/view/2932">Bessel polynomials by context-free grammars</a> (Polinomios de Bessel mediante gramáticas independientes del contexto), Bistua, Univ. de Pamplona (Colombia, 2024) Vol 22, No. 2. See p. 3.

%H Jonas Wahl, <a href="https://arxiv.org/abs/2009.08181">Traces on diagram algebras II: Centralizer algebras of easy groups and new variations of the Young graph</a>, arXiv:2009.08181 [math.RT], 2020.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ModifiedSphericalBesselFunctionoftheSecondKind.html">Modified Spherical Bessel Function of the Second Kind</a>

%H <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>

%F a(n, k) = (n+k)!/(2^k*(n-k)!*k!) (see Grosswald and Riordan). - _Ralf Stephan_, Apr 20 2004

%F 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_

%F a(n, m) = A001497(n, n-m) = A001147(m)*binomial(n+m, 2*m) for n >= m >= 0, otherwise 0.

%F 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).

%F 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

%F 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

%F 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

%F T(n, k) = Pochhammer(n+1,k)*binomial(n,k)/2^k = A113025(n,k)/2^k. - _Peter Luschny_, May 11 2018

%e The triangle a(n, k), n >= 0, k = 0..n, begins:

%e 1

%e 1 1

%e 1 3 3

%e 1 6 15 15

%e 1 10 45 105 105

%e 1 15 105 420 945 945

%e 1 21 210 1260 4725 10395 10395

%e 1 28 378 3150 17325 62370 135135 135135

%e 1 36 630 6930 51975 270270 945945 2027025 2027025

%e 1 45 990 13860 135135 945945 4729725 16216200 34459425 34459425

%e ...

%e y_0(x) = 1

%e y_1(x) = x + 1

%e y_2(x) = 3*x^2 + 3*x + 1

%e y_3(x) = 15*x^3 + 15*x^2 + 6*x + 1

%e y_4(x) = 105*x^4 + 105*x^3 + 45*x^2 + 10*x + 1

%e y_5(x) = 945*x^5 + 945*x^4 + 420*x^3 + 105*x^2 + 15*x + 1

%e 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

%p Bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end; # explicit Bessel polynomials

%p 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

%p bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end;

%p 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;

%p # Alternative:

%p T := (n,k) -> pochhammer(n+1,k)*binomial(n,k)/2^k:

%p for n from 0 to 9 do seq(T(n,k), k=0..n) od; # _Peter Luschny_, May 11 2018

%p T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)

%p else (n - k + 1)* T(n, k - 1) + T(n - 1, k) fi fi end:

%p for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # _Peter Luschny_, Oct 02 2023

%t 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 *)

%o (PARI) {T(n,k)=if(k<0||k>n, 0, binomial(n, k)*(n+k)!/2^k/n!)} /* _Michael Somos_, Oct 03 2006 */

%o (PARI)

%o A001497_ser(N,t='t) = {

%o my(x='x+O('x^(N+2)));

%o serlaplace(deriv(exp((1-sqrt(1-2*t*x))/t),'x));

%o };

%o concat(apply(Vecrev, Vec(A001497_ser(9)))) \\ _Gheorghe Coserea_, Dec 27 2017

%o (Haskell)

%o a001498 n k = a001498_tabl !! n !! k

%o a001498_row n = a001498_tabl !! n

%o a001498_tabl = map reverse a001497_tabl

%o -- _Reinhard Zumkeller_, Jul 11 2014

%o (Magma) /* 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

%Y Cf. A001497 (same triangle but rows read in reverse order). Other versions of this same triangle are given in A144331, A144299, A111924 and A100861.

%Y Columns from left edge include A000217, A050534.

%Y Columns 1-6 from right edge are A001147, A001879, A000457, A001880, A001881, A038121.

%Y 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.

%Y Cf. A245066 (central terms). A113025 (y_n(2*x)).

%K nonn,tabl,nice,easy

%O 0,5

%A _N. J. A. Sloane_