0,4

The (reverse) Bessel polynomials P(n,x):=sum(a(n,m)*x^m,m=0..n), the row polynomials, called Theta_n(x) in the Grosswald reference, solve x*diff(P(n,x),x$2)-2*(x+n)*diff(P(n,x),x)+2*n*P(n,x)) = 0.

With the related Sheffer associated polynomials defined by Carlitz as

B(0,x) = 1

B(1,x) = x

B(2,x) = x + x^2

B(3,x) = 3 x + 3 x^2 + x^3

B(4,x) = 15 x + 15 x^2 + 6 x^3 + x^4

... (see Mathworld reference), then P(n,x) = 2^n * B(n,x/2) are the Sheffer polynomials described in A119274. - Tom Copeland (tcjpn(AT)msn.com), Feb 10 2008

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

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

Eric Weisstein's World of Mathematics, Bessel Polynomial

Index entries for sequences related to Bessel functions or polynomials

a(n, m)=(2*n-m)!/(m!*(n-m)!*2^(n-m)) if n >= m >= 0 else 0 (from Grosswald, p. 7).

a(n, m)= 0, n<m; a(n, -1) := 0; a(0, 0)= 1; a(n, m) = (2*n-m-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 0 (from Grosswald p. 23, (19)).

E.g.f. for m-th column: ((1-sqrt(1-2*x))^m)/(m!*sqrt(1-2*x)).

G.f.: 1/(1-xy-x/(1-xy-2x/(1-xy-3x/(1-xy-4x/(1-.... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Jan 29 2009]

f := proc(n) option remember; if n <=1 then (1+x)^n else expand((2*n-1)*x*f(n-1)+f(n-2)); fi; end;

(PARI) T(k, n) = if(n>k||k<0||n<0, 0, (2*k-n)!/(n!*(k-n)!*2^(k-n))) (from R. Stephan)

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

Reflected version of A001498 which is considered the main entry.

Other versions of this same triangle are given in A144299, A111924 and A100861.

Row sums give A001515. a(n, 0)= A001147(n) (double factorials).

Sequence in context: A039797 A143171 A112292 this_sequence A123244 A105599 A106210

Adjacent sequences: A001494 A001495 A001496 this_sequence A001498 A001499 A001500

nonn,tabl,nice

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