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A001497
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Triangle of coefficients of Bessel polynomials (exponents in decreasing order).
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33
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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
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OFFSET
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0,4
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COMMENTS
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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)]. [From Paul Barry, Jul 27 2010]
Contribution from Vladimir, 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)
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REFERENCES
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E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
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.
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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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]
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MAPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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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 A106210
Adjacent sequences: A001494 A001495 A001496 * A001498 A001499 A001500
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KEYWORD
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nonn,tabl,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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