A001518 Bessel polynomial y_n(3).
1, 4, 37, 559, 11776, 318511, 10522639, 410701432, 18492087079, 943507142461, 53798399207356, 3390242657205889, 233980541746413697, 17551930873638233164, 1421940381306443299981, 123726365104534205331511, 11507973895102987539130504
Offset: 0
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
Programs
-
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 *) -
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
Formula
y_n(x) = Sum_{k=0..n} (n+k)!*(x/2)^k/((n-k)!*k!).
D-finite with recurrence 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)