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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). 36
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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(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.

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-th 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} [From Leonid Bedratyuk, Aug 06 2010]

REFERENCES

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.

Alexander W. Boldyreff, Decomposition of Rational Fractions into Partial Fractions, Nat. Math. Mag. 17 (6) (1943), 261-267; coefficients (m)N(r).

Frink, O. and H. L. Krall, A new class of orthogonal polynomials, Trans. Amer. Math. Soc. 65, 100-115, 1945

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

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

LINKS

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.

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

S.-M. Ma, T. Mansour, M. Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169, 2013

L. A. Székelys, P. L. Erdős 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

FORMULA

a(n, k) = (n+k)!/(2^k*(n-k)!*k!) (see Grosswald). - 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) [ 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).

Row polynomials are given by D^n(exp(x)) evaluated at x = 0, where D is the operator 1/(1-x*t)*d/dx. - 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

EXAMPLE

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.

MAPLE

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;

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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: A094040 A039798 A193560 * A240439 A199034 A138464

Adjacent sequences:  A001495 A001496 A001497 * A001499 A001500 A001501

KEYWORD

nonn,tabl,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 23 19:18 EDT 2014. Contains 240946 sequences.