Search: id:A001497
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%I A001497
%S A001497 1,1,1,3,3,1,15,15,6,1,105,105,45,10,1,945,945,420,105,15,1,10395,10395,
%T A001497 4725,1260,210,21,1,135135,135135,62370,17325,3150,378,28,1,2027025,
%U A001497 2027025,945945,270270,51975,6930,630,36,1,34459425,34459425
%N A001497 Triangle of coefficients of Bessel polynomials (exponents in decreasing 
               order).
%C A001497 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.
%C A001497 With the related Sheffer associated polynomials defined by Carlitz as
%C A001497 B(0,x) = 1
%C A001497 B(1,x) = x
%C A001497 B(2,x) = x + x^2
%C A001497 B(3,x) = 3 x + 3 x^2 + x^3
%C A001497 B(4,x) = 15 x + 15 x^2 + 6 x^3 + x^4
%C A001497 ... (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
%D A001497 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 A001497 E Grosswald, Bessel Polynomials, Lecture Notes Math. vol. 698 1978 p. 
               18.
%D A001497 B. Leclerc, Powers of staircase Schur functions and symmetric analogues 
               of Bessel polynomials, Discrete Math., 153 (1996), 213-227.
%D A001497 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%H A001497 T. D. Noe, <a href="b001497.txt">Rows n=0..50 of triangle, flattened</
               a>
%H A001497 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 A001497 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BesselPolynomial.html">Bessel Polynomial</a>
%H A001497 <a href="Sindx_Be.html#Bessel">Index entries for sequences related to 
               Bessel functions or polynomials</a>
%F A001497 a(n, m)=(2*n-m)!/(m!*(n-m)!*2^(n-m)) if n >= m >= 0 else 0 (from Grosswald, 
               p. 7).
%F A001497 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)).
%F A001497 E.g.f. for m-th column: ((1-sqrt(1-2*x))^m)/(m!*sqrt(1-2*x)).
%F A001497 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]
%p A001497 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;
%o A001497 (PARI) T(k, n) = if(n>k||k<0||n<0,0,(2*k-n)!/(n!*(k-n)!*2^(k-n))) (from 
               R. Stephan)
%o A001497 (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 */
%Y A001497 Reflected version of A001498 which is considered the main entry.
%Y A001497 Other versions of this same triangle are given in A144299, A111924 and 
               A100861.
%Y A001497 Row sums give A001515. a(n, 0)= A001147(n) (double factorials).
%Y A001497 Sequence in context: A039797 A143171 A112292 this_sequence A123244 A105599 
               A106210
%Y A001497 Adjacent sequences: A001494 A001495 A001496 this_sequence A001498 A001499 
               A001500
%K A001497 nonn,tabl,nice
%O A001497 0,4
%A A001497 N. J. A. Sloane (njas(AT)research.att.com).

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