A065919 Bessel polynomial y_n(4).

1, 5, 61, 1225, 34361, 1238221, 54516085, 2836074641, 170218994545, 11577727703701, 880077524475821, 73938089783672665, 6803184337622361001, 680392371852019772765, 73489179344355757819621, 8525425196317119926848801, 1057226213522667226687070945

Offset

0,2

Comments

Main diagonal of A143411. - Peter Bala, Aug 14 2008

References

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

Links

Harry J. Smith, Table of n, a(n) for n = 0..100

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

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!).

From Peter Bala, Aug 14 2008: (Start)

Recurrence relation: a(0) = 1, a(1) = 5, a(n) = 4*(2*n-1)*a(n-1) + a(n-2) for n >= 2. Sequence A143412(n) satisfies the same recurrence relation.

1/sqrt(e) = 1 - 2*Sum_{n = 0..inf} (-1)^n/(a(n)*a(n+1)) = 1 - 2*( 1/(1*5) - 1/(5*61) + 1/(61*1225) - ... ). (End)

G.f.: 1/Q(0), where Q(k)= 1 - x - 4*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013

a(n) = exp(1/4)/sqrt(2*Pi)*BesselK(n+1/2,1/4). - Gerry Martens, Jul 22 2015

a(n) ~ 2^(3*n+1/2) * n^n / exp(n-1/4). - Vaclav Kotesovec, Jul 22 2015

From Peter Bala, Apr 12 2017: (Start)

a(n) = 1/n!*Integral_{x = 0..inf} x^n*(1 + 2*x)^n dx.

E.g.f.: d/dx( exp(x*c(2*x)) ) = 1 + 5*x + 61*x^2/2! + 1225*x^3/3! + ..., where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

G.f.: (1/(1-x))*hypergeometric2f0(1,1/2; - ; 8*x/(1-x)^2). - G. C. Greubel, Aug 16 2017

a(n) = 2^n*KummerU(-n, -2*n, 1/2). - Peter Luschny, May 10 2022

Maple

seq(simplify(2^n*KummerU(-n, -2*n, 1/2)), n=0..16); # Peter Luschny, May 10 2022

Mathematica

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

Prog

(PARI) for (n=0, 100, if (n>1, a=4*(2*n - 1)*a1 + a2; a2=a1; a1=a, if (n, a=a1=5, a=a2=1)); write("b065919.txt", n, " ", a) ) \\ Harry J. Smith, Nov 04 2009
(PARI) a(n) = sum(k=0, n, (n+k)!*2^k/((n-k)!*k!) ); \\ Joerg Arndt, May 17 2013
(Magma)
A065919:= func< n | (&+[Binomial(n, k)*Factorial(n+k)*2^k/Factorial(n): k in [0..n]]) >;
[A065919(n): n in [0..30]]; // G. C. Greubel, Oct 05 2023
(SageMath)
def A065919(n): return sum(binomial(n, k)*factorial(n+k)*2^k/factorial(n) for k in range(n+1))
[A065919(n) for n in range(31)] # G. C. Greubel, Oct 05 2023

Crossrefs

Cf. A001515, A001517, A001518.

Cf. A143411 (main diagonal), A143412.

Polynomial coefficients are in A001498.

Sequence in context: A217820 A217821 A009825 * A196125 A345103 A096537

Adjacent sequences: A065916 A065917 A065918 * A065920 A065921 A065922

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 08 2001

Status

approved