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A001881 Coefficients of Bessel polynomials y_n (x).
(Formerly M5116 N2217)
7

%I M5116 N2217 #50 Sep 08 2022 08:44:29

%S 1,21,378,6930,135135,2837835,64324260,1571349780,41247931725,

%T 1159525191825,34785755754750,1109981842719750,37554385678684875,

%U 1343291487737574375,50661278966102805000,2009564065655411265000,83648104232906493905625,3646073249210806587298125

%N Coefficients of Bessel polynomials y_n (x).

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H G. C. Greubel, <a href="/A001881/b001881.txt">Table of n, a(n) for n = 5..400</a> (terms 5..100 from T. D. Noe)

%H Selden Crary, Richard Diehl Martinez, Michael Saunders, <a href="https://arxiv.org/abs/1707.00705">The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters</a>, arXiv:1707.00705 [stat.ME], 2017, Table 1.

%H J. Riordan, <a href="/A001820/a001820.pdf">Notes to N. J. A. Sloane, Jul. 1968</a>

%H <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>

%F a(n) = (2n-5)!/( 5!*(n-5)!*2^(n-5) ).

%F a(n) = binomial(n-3,2)*(2*n-5)!!/5!!, n >= 5, with (2*n-5)!! = A001147(n-2).

%F E.g.f.: x*(1 + 3*x/2)/(1 - 2*x)^(9/2), with offset 1. - _G. C. Greubel_, Aug 13 2017

%F G.f.: t^5 * hypergeometric2F0(3, 7/2; -; 2*t) = t^5 + 21*t^6 + .... - _G. C. Greubel_, Aug 16 2017

%t With[{nn = 50}, CoefficientList[Series[x*(1 + 3*x/2)/(1 - 2*x)^(9/2), {x, 0, nn}], x]*Range[0, nn]!] (* _G. C. Greubel_, Aug 13 2017 *)

%o (PARI) x='x+O('x^50); Vec(serlaplace(x*(1 + 3*x/2)/(1 - 2*x)^(9/2))) \\ _G. C. Greubel_, Aug 13 2017

%o (Magma) [Factorial(2*n-5)/(120*Factorial(n-5)*2^(n-5) ): n in [5..30]]; // _Vincenzo Librandi_, Aug 14 2017

%Y See A001518.

%Y (1/4) the coefficient of x^2 of polynomials in A098503.

%Y Column 5 of triangle A001497.

%Y Third column (m=2) of Laguerre-Sonin a=1/2 triangle A130757.

%K nonn,easy

%O 5,2

%A _N. J. A. Sloane_, _Simon Plouffe_

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Last modified March 29 05:28 EDT 2024. Contains 371264 sequences. (Running on oeis4.)