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A001497 Triangle of coefficients of Bessel polynomials (exponents in decreasing order). 33
1, 1, 1, 3, 3, 1, 15, 15, 6, 1, 105, 105, 45, 10, 1, 945, 945, 420, 105, 15, 1, 10395, 10395, 4725, 1260, 210, 21, 1, 135135, 135135, 62370, 17325, 3150, 378, 28, 1, 2027025, 2027025, 945945, 270270, 51975, 6930, 630, 36, 1, 34459425, 34459425 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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, Feb 10 2008

Exponential Riordan array [1/sqrt(1-2x),1-sqrt(1-2x)]. [Paul Barry, Jul 27 2010]

From Vladimir Kruchinin, Mar 18 2011: (Start)

For B(n,k){...} the Bell polynomial of the second kind we have

B(n,k){f',f'',f''',...}=T(n+1,k+1)*(1-2*x)^(k/2-n), where f(x)=(1-sqrt(1-2*x)

The expansions of the first few rows are:

1/sqrt(1-2*x);

1/(1-2*x)^(3/2), 1/(1-2*x);

3/(1-2*x)^(5/2), 3/(1-2*x)^2, 1/(1-2*x)^(3/2);

15/(1-2*x)^(7/2),15/(1-2*x)^3, 6/(1-2*x)^(5/2), 1/(1-2*x)^2]; (End)

REFERENCES

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

Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From N. J. A. Sloane, Sep 15 2012

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

LINKS

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

E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.

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

Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.

W. Mlotkowski, A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.

Eric Weisstein's World of Mathematics, Bessel Polynomial

Index entries for sequences related to Bessel functions or polynomials

FORMULA

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, Jan 29 2009]

T(n,k)=if(k<=n, C(2n-k,2(n-k))*(2(n-k)-1)!!,0)=if(k<=n, C(2n-k,2(n-k))*A001147(n-k),0) [Paul Barry, Mar 18 2011]

Row polynomials for n>=1 are given by 1/t*D^n(exp(x*t)) evaluated at x = 0, where D is the operator 1/(1-x)*d/dx. - Peter Bala, Nov 25 2011

EXAMPLE

Triangle begins

1,

1, 1,

3, 3, 1,

15, 15, 6, 1,

105, 105, 45, 10, 1,

945, 945, 420, 105, 15, 1,

10395, 10395, 4725, 1260, 210, 21, 1,

135135, 135135, 62370, 17325, 3150, 378, 28, 1,

2027025, 2027025, 945945, 270270, 51975, 6930, 630, 36, 1

Production matrix begins

1, 1,

2, 2, 1,

6, 6, 3, 1,

24, 24, 12, 4, 1,

120, 120, 60, 20, 5, 1,

720, 720, 360, 120, 30, 6, 1,

5040, 5040, 2520, 840, 210, 42, 7, 1,

40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1,

362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1

This is the exponential Riordan array A094587, or [1/(1-x),x], beheaded.

[Paul Barry, Mar 18 2011]

MAPLE

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;

MATHEMATICA

m = 9; Flatten[ Table[(n + k)!/(2^k*k!*(n - k)!), {n, 0, m}, {k, n, 0, -1}]] (* Jean-François Alcover, Sep 20 2011 *)

y[n_, x_] := Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n-1/2, 1/x]; t[n_, k_] := Coefficient[y[n, x], x, k]; Table[t[n, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 01 2013 *)

PROG

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

CROSSREFS

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).

Cf. A104556 (matrix inverse).

Sequence in context: A039797 A143171 A112292 * A123244 A105599 A239895

Adjacent sequences:  A001494 A001495 A001496 * A001498 A001499 A001500

KEYWORD

nonn,tabl,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 19 17:12 EDT 2014. Contains 240762 sequences.