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A001518 Bessel polynomial y_n(3).
(Formerly M3669 N1495)
18
1, 4, 37, 559, 11776, 318511, 10522639, 410701432, 18492087079, 943507142461, 53798399207356, 3390242657205889, 233980541746413697, 17551930873638233164, 1421940381306443299981, 123726365104534205331511, 11507973895102987539130504 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

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

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

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

LINKS

Gheorghe Coserea and T. D. Noe, Table of n, a(n) for n = 0..200 (terms up to n=100 by T. D. Noe)

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

Simon Plouffe, Approximations of generating functions and a few conjectures, arXiv:0911.4975 [math.NT], 2009.

J. Riordan, Letter to N. J. A. Sloane, Jul. 1968

N. J. A. Sloane, Letter to J. Riordan, Nov. 1970

Index entries for sequences related to Bessel functions or polynomials

FORMULA

y_n(x) = Sum_{k=0..n} (n+k)!*(x/2)^k/((n-k)!*k!).

a(n) = 3(2n-1)*a(n-1) + a(n-2). - T. D. Noe, Oct 26 2006

G.f.: 1/Q(0), where Q(k)= 1 - x - 3*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013

a(n) = exp(1/3)*sqrt(2/(3*Pi))*BesselK(1/2+n,1/3). - Gerry Martens, Jul 22 2015

a(n) ~ sqrt(2) * 6^n * n^n / exp(n-1/3). - Vaclav Kotesovec, Jul 22 2015

E.g.f.: exp(1/3 - 1/3*(1-6*x)^(1/2)) / (1-6*x)^(1/2). (formula due to B. Salvy, see Plouffe link) - Gheorghe Coserea, Aug 06 2015

From G. C. Greubel, Aug 16 2017: (Start)

a(n) = (1/2)_{n} * 6^n * hypergeometric1f1(-n; -2*n; 2/3).

G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; 6*t/(1-t)^2). (End)

MAPLE

f:= gfun:-rectoproc({a(n)=3*(2*n-1)*a(n-1)+a(n-2), a(0)=1, a(1)=4}, a(n), remember):

map(f, [$0..60]); # Robert Israel, Aug 06 2015

MATHEMATICA

Table[Sum[(n+k)!*3^k/(2^k*(n-k)!*k!), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 22 2015 *)

PROG

(PARI) x='x+O('x^33); Vec(serlaplace(exp(1/3 - 1/3 * (1-6*x)^(1/2)) / (1-6*x)^(1/2))) \\ Gheorghe Coserea, Aug 04 2015

CROSSREFS

Cf. A001515, A001517.

Polynomial coefficients are in A001498.

Sequence in context: A316877 A277638 A121080 * A185082 A259822 A036245

Adjacent sequences:  A001515 A001516 A001517 * A001519 A001520 A001521

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 17 10:59 EST 2019. Contains 319218 sequences. (Running on oeis4.)