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A122848
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Exponential Riordan array (1, x(1+x/2)).
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13
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1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 0, 3, 6, 1, 0, 0, 0, 15, 10, 1, 0, 0, 0, 15, 45, 15, 1, 0, 0, 0, 0, 105, 105, 21, 1, 0, 0, 0, 0, 105, 420, 210, 28, 1, 0, 0, 0, 0, 0, 945, 1260, 378, 36, 1, 0, 0, 0, 0, 0, 945, 4725, 3150, 630, 45, 1, 0, 0, 0, 0, 0, 0, 10395, 17325, 6930, 990, 55, 1, 0, 0
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OFFSET
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0,9
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COMMENTS
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Entries are Bessel polynomial coefficients. Row sums are A000085. Diagonal sums are A122849. Inverse is A122850. Product of A007318 and A122848 gives A100862.
T(n,k) is the number of self inverse permutations of {1,2,...,n} having exactly k cycles. - Geoffrey Critzer, May 08 2012
Bessel numbers of the second kind. For relations to the Hermite polynomials and the Catalan (A033184 and A009766) and Fibonacci (A011973, A098925, and A092865) matrices, see Yang and Qiao. - Tom Copeland, Dec 18 2013.
Also the inverse Bell transform of the double factorial of odd numbers Product_{k= 0..n-1} (2*k+1) (A001147). For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015
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LINKS
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G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
P. Bala, Generalized Dobinski formulas
Richell O. Celeste, Roberto B. Corcino, Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients. Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
H. Han, S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, Eur. J. Combinat. 29 (7) (2008) 1544-1554. [From R. J. Mathar, Mar 20 2009]
S. Yang and Z. Qiao, The Bessel Numbers and Bessel Matrices, Journal of Mathematical Research & Exposition, July, 2011, Vol. 31, No. 4, pp. 627-636. [From Tom Copeland, Dec 18 2013]
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FORMULA
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Number triangle T(n,k) = k!*C(n,k)/((2k-n)!*2^(n-k)).
T(n,k) = A001498(k,n-k). - Michael Somos, Oct 03 2006
E.g.f.: exp(y(x+x^2/2)). - Geoffrey Critzer, May 08 2012
Triangle equals the matrix product A008275*A039755. Equivalently, the n-th row polynomial R(n,x) is given by the Type B Dobinski formula R(n,x) = exp(-x/2)*Sum_{k>=0} P(n,2*k+1)*(x/2)^k/k!, where P(n,x) = x*(x-1)*...*(x-n+1) denotes the falling factorial polynomial. Cf. A113278. - Peter Bala, Jun 23 2014
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EXAMPLE
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Triangle begins
1;
0, 1;
0, 1, 1;
0, 0, 3, 1;
0, 0, 3, 6, 1;
0, 0, 0, 15, 10, 1;
0, 0, 0, 15, 45, 15, 1;
0, 0, 0, 0, 105, 105, 21, 1;
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MAPLE
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# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n<2, 1, 0), 9); # Peter Luschny, Jan 27 2016
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MATHEMATICA
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t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten
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PROG
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(PARI) {T(n, k)=if(2*k<n|k>n, 0, n!/(2*k-n)!/(n-k)!*2^(k-n))} /* Michael Somos, Oct 03 2006 */
(Sage)
# The function inverse_bell_transform is defined in A265605.
multifact_2_1 = lambda n: prod(2*k + 1 for k in (0..n-1))
inverse_bell_matrix(multifact_2_1, 9) # Peter Luschny, Dec 31 2015
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CROSSREFS
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Cf. A001497, A008275, A039755, A049403, A111924, A113278.
Sequence in context: A170846 A085604 A144357 * A272481 A054548 A059202
Adjacent sequences: A122845 A122846 A122847 * A122849 A122850 A122851
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Paul Barry, Sep 14 2006
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STATUS
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approved
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