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 A001497 Triangle of coefficients of Bessel polynomials (exponents in decreasing order). 43
 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, 16216200, 4729725, 945945, 135135, 13860, 990, 45, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The (reverse) Bessel polynomials P(n,x):=Sum_{m=0..n} a(n,m)*x^m, the row polynomials, called Theta_n(x) in the Grosswald reference, solve x*(d^2/dx^2)P(n,x) - 2*(x+n)*(d/dx)P(n,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) Also the Bell transform of A001147 (whithout column 0 which is 1,0,0,...). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016 Antidiagonals of A099174 are rows of this entry. Dividing each diagonal by its first element generates A054142. - Tom Copeland, Oct 04 2016 The row polynomials p_n(x) of A107102 are (-1)^n B_n(1-x), where B_n(x) are the modified Carlitz-Bessel polynomials above, e.g., (-1)^2 B_2(1-x) = (1-x) + (1-x)^2 = 2 - 3 x + x^2 = p_2(x). - Tom Copeland, Oct 10 2016 a(n-1,m-1) counts rooted unordered binary forests with n labeled leaves and m roots. - David desJardins, Feb 23 2019 From Jianing Song, Nov 29 2021: (Start) The polynomials P_n(x) = Sum_{k=0..n} T(n,k)*x^k satisfy: P_n(x) - (d/dx)P_n(x) = x*P_{n-1}(x) for n >= 1. {P(n,x)} are related to the Fourier transform of 1/(1+x^2)^(n+1) and x/(1+x^2)^(n+2): (i) For n >= 0, real number t, we have Integral_{x=-oo..oo} exp(-i*t*x)/(1+x^2)^(n+1) dx = Pi/(2^n*n!) * P_n(|t|) * exp(-|t|); (ii) For n >= 0, real number t, we have Integral_{x=-oo..oo} x*exp(-i*t*x)/(1+x^2)^(n+2) dx = Pi/(2^(n+1)*(n+1)!) * ((-t)*P_n(-|t|)) * exp(-|t|). (End) REFERENCES J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77. LINKS T. D. Noe, Rows n = 0..50 of triangle, flattened Peter Bala, Generalized Dobinski formulas Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, chapter 8. E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122. O. Frink 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. Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) #09.8.3. Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. 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, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3. 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 W. Mlotkowski and A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408. Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019. Feng Qi and Bai-Ni Guo, "Some Properties and Generalizations of the Catalan, Fuss, and Fuss-Catalan Numbers", Mathematical Analysis and Applications : Selected Topics (2018), Wiley, Ch. 5, 101-133. Feng Qi, X.-T. Shi and F.-F. Liu, Several formulas for special values of the Bell polynomials of the second kind and applications, Preprint 2015. Alexander Stoimenow, On the number of chord diagrams, Discr. Math. 218 (2000), 209-233. Lemma 2.2. Eric Weisstein's World of Mathematics, Bessel Polynomial 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 >= 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). - 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 The matrix product A039683*A008277 gives a signed version of this triangle. Dobinski-type formula for the row polynomials: R(n,x) = (-1)^n*exp(x)*Sum_{k = 0..inf} k*(k-2)*(k-4)*...*(k-2*(n-1))*(-x)^k/k!. Cf. A122850. - Peter Bala, Jun 23 2014 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; row := n -> seq(coeff(f(n), x, n - k), k = 0..n): seq(row(n), n = 0..9); 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))) /* 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 */ (Haskell) a001497 n k = a001497_tabl !! n !! k a001497_row n = a001497_tabl !! n a001497_tabl = [1] : f [1] 1 where    f xs z = ys : f ys (z + 2) where      ys = zipWith (+) ([0] ++ xs) (zipWith (*) [z, z-1 ..] (xs ++ [0])) -- Reinhard Zumkeller, Jul 11 2014 (MAGMA) /* As triangle */ [[Factorial(2*n-k)/(Factorial(k)*Factorial(n-k)*2^(n-k)): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 12 2015 (Sage) # uses[bell_matrix from A264428] # Adds a column 1, 0, 0, 0, ... at the left side of the triangle. bell_matrix(lambda n: A001147(n), 9) # Peter Luschny, Jan 19 2016 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). A039683, A122850. Cf. A245066 (central terms). Cf. A054142, A099174, A107102. Sequence in context: A039797 A143171 A112292 * A123244 A105599 A239895 Adjacent sequences:  A001494 A001495 A001496 * A001498 A001499 A001500 KEYWORD nonn,tabl,nice AUTHOR STATUS approved

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Last modified June 27 08:15 EDT 2022. Contains 354891 sequences. (Running on oeis4.)