0,5

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 (pbarry(AT)wit.ie), 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 (pbarry(AT)wit.ie), Aug 17 2005

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

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.

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.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 2009: (Start)

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.

(End)

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.

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]

E. Grosswald, Bessel Polynomials, Lecture Notes Math. vol. 698 1978 p. 18.

B. Leclerc, Powers of staircase Schur functions and symmetric analogues of Bessel polynomials, Discrete Math., 153 (1996), 213-227.

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

T. D. Noe, Rows n=0..50 of triangle, flattened

David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

A. Burstein and T. Mansour, Words restricted by patterns with at most 2 distinct letters.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

W. Lang, First ten rows.

L. A. Sz\'ekely, P. L. Erd\"os and M. A. Steel, The combinatorics of evolutionary trees

Eric Weisstein's World of Mathematics, Modified Spherical Bessel Function of the Second Kind

Index entries for sequences related to Bessel functions or polynomials

a(n, k) = (n+k)!/(2^k*(n-k)!*k!) (see Grosswald). - R. 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) [ Leonard Smiley (smiley(AT)math.uaa.alaska.edu) ]

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).

Triangle 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

...

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). W. Lang, Sep 14 2007.

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;

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

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.

Sequence in context: A143389 A094040 A039798 this_sequence A138464 A117279 A049323

Adjacent sequences: A001495 A001496 A001497 this_sequence A001499 A001500 A001501

nonn,tabl,nice

N. J. A. Sloane (njas(AT)research.att.com).