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 T(n,k) has factorization [C(n,k)][C(k,n-k)]Diag((2n-1)!!) The triangle T(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 T(n,k)=A100861(n+k,k). Related to the Morgan-Voyce polynomials by T(n,k)=(2k-1)!!*A085478(n,k). - Paul Barry (pbarry(AT)wit.ie), Aug 17 2005

Related to Hermite polynomials by T(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) solve (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.

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.

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.

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

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.

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

W. Lang, First ten rows.

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 else 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).

B(r, s) = (r+s)!/[2^s*(r-s)!*s! ]. - R. Stephan, Apr 20 2004

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 combinatorics: 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 */

Columns from left edge include A000217, A050534.

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

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. A001497.

Adjacent sequences: A001495 A001496 A001497 this_sequence A001499 A001500 A001501

Sequence in context: A110640 A094040 A039798 this_sequence A117279 A049323 A084144

nonn,tabl,nice

njas