%I A001498
%S A001498 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 A001498 1260,4725,10395,10395,1,28,378,3150,17325,62370,135135,135135,1,36,630,
%U A001498 6930,51975,270270,945945,2027025,2027025,1,45,990,13860,135135
%N A001498 Triangle a(n,k) (n>=0, 0<=k<=n) of coefficients of Bessel polynomials 
               y_n(x) (exponents in increasing order).
%C A001498 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 A001498 Also called Bessel numbers of first kind.
%C A001498 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 (pbarry(AT)wit.ie), May 21 2005
%C A001498 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 (pbarry(AT)wit.ie), Aug 17 2005
%C A001498 Related to Hermite polynomials by a(n,k)=(-1)^k*A060821(n+k,n-k)/2^n; 
               - Paul Barry (pbarry(AT)wit.ie), Aug 28 2005
%C A001498 The row polynomials, the Bessel polynomials y(n,x):=sum(a(n,m)*x^m,m=0..n) 
               (called y_{n}(x) in the Grosswald reference) satisfy (x^2)*diff(y(n,
               x),x$2)+2*(x+1)*diff(y(n,x),x)-n*(n+1)*y(n,x)) = 0.
%C A001498 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. W. Lang, Sep 
               14 2007.
%C A001498 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 
               2009: (Start)
%C A001498 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.
%C A001498 (End)
%D A001498 F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, 
               in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, 
               Springer 1922, pp. 24-48.
%D A001498 Frink, O. and H. L. Krall, A new class of orthogonal polynomials, Trans. 
               Amer. Math. Soc. 65,100-115, 1945 [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), 
               Feb 15 2009]
%D A001498 E. Grosswald, Bessel Polynomials, Lecture Notes Math. vol. 698 1978 p. 
               18.
%D A001498 B. Leclerc, Powers of staircase Schur functions and symmetric analogues 
               of Bessel polynomials, Discrete Math., 153 (1996), 213-227.
%D A001498 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%H A001498 T. D. Noe, <a href="b001498.txt">Rows n=0..50 of triangle, flattened</
               a>
%H A001498 David Applegate and N. J. A. Sloane, <a href="http://arxiv.org/abs/0907.0513">
               The Gift Exchange Problem</a> (arXiv:0907.0513, 2009)
%H A001498 A. Burstein and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0110056">
               Words restricted by patterns with at most 2 distinct letters</a>.
%H A001498 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               On generalizations of Stirling number triangles</a>, J. Integer Seqs., 
               Vol. 3 (2000), #00.2.4.
%H A001498 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A001498.text">
               First ten rows. </a>
%H A001498 L. A. Sz\'ekely, P. L. Erd\"os and M. A. Steel, <a href="http://www.mat.univie.ac.at/
               ~slc/opapers/s28szekely.html">The combinatorics of evolutionary trees</
               a>
%H A001498 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ModifiedSphericalBesselFunctionoftheSecondKind.html">Modified Spherical 
               Bessel Function of the Second Kind</a>
%H A001498 <a href="Sindx_Be.html#Bessel">Index entries for sequences related to 
               Bessel functions or polynomials</a>
%F A001498 a(n, k) = (n+k)!/(2^k*(n-k)!*k!) (see Grosswald). - R. Stephan, Apr 20 
               2004
%F A001498 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) [ Leonard Smiley (smiley(AT)math.uaa.alaska.edu) 
               ]
%F A001498 a(n, m)= A001497(n, n-m) = A001147(m)*binomial(n+m, 2*m) for n >= m >
               = 0, otherwise 0.
%F A001498 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).
%e A001498 Triangle begins:
%e A001498 1
%e A001498 1 1
%e A001498 1 3 3
%e A001498 1 6 15 15
%e A001498 1 10 45 105 105
%e A001498 1 15 105 420 945 945
%e A001498 1 21 210 1260 4725 10395 10395
%e A001498 1 28 378 3150 17325 62370 135135 135135
%e A001498 1 36 630 6930 51975 270270 945945 2027025 2027025
%e A001498 1 45 990 13860 135135 945945 4729725 16216200 34459425 34459425
%e A001498 ...
%e A001498 y_0(x) = 1
%e A001498 y_1(x) = x + 1
%e A001498 y_2(x) = 3*x^2 + 3*x + 1
%e A001498 y_3(x) = 15*x^3 + 15*x^2 + 6*x + 1
%e A001498 y_4(x) = 105*x^4 + 105*x^3 + 45*x^2 + 10*x + 1
%e A001498 y_5(x) = 945*x^5 + 945*x^4 + 420*x^3 + 105*x^2 + 15*x + 1
%e A001498 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). W. Lang, Sep 14 2007.
%p A001498 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 A001498 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 A001498 bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); 
               end;
%p A001498 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;
%o A001498 (PARI) {T(n,k)=if(k<0|k>n, 0, binomial(n, k)*(n+k)!/2^k/n!)} /* Michael 
               Somos Oct 03 2006 */
%Y A001498 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 A001498 Columns from left edge include A000217, A050534.
%Y A001498 Columns 1-6 from right edge are A001147, A001879, A000457, A001880, A001881, 
               A038121.
%Y A001498 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 A001498 Sequence in context: A143389 A094040 A039798 this_sequence A138464 A117279 
               A049323
%Y A001498 Adjacent sequences: A001495 A001496 A001497 this_sequence A001499 A001500 
               A001501
%K A001498 nonn,tabl,nice
%O A001498 0,5
%A A001498 N. J. A. Sloane (njas(AT)research.att.com).