A065947 - OEIS

%I #18 Aug 17 2017 06:29:17

%S 0,0,6,300,13320,620130,31406550,1743174216,105889417200,

%T 7010411889690,503353562247360,39003404559533700,3246506259033473436,

%U 289042023964190515200,27418894569798460848210,2761554229456140638184840,294364593823858690215256200

%N Bessel polynomial {y_n}''(3).

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

%H G. C. Greubel, <a href="/A065947/b065947.txt">Table of n, a(n) for n = 0..340</a>

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

%F From _G. C. Greubel_, Aug 14 2017: (Start)

%F a(n) = 4*n*(n - 1)*(1/2)_{n}*6^(n - 2)*hypergeometric1F1[(2-n, -2*n, 2/3), where (a)_{n} is the Pochhammer symbol.

%F E.g.f.: (-1/81)*(1 - 6*x)^(-5/2)*((171*x^2 - 90*x + 8)*sqrt(1 - 6*x) + (54*x^3 - 648*x^2 + 114*x - 8))*exp((1 - sqrt(1 - 6*x))/3). (End)

%F G.f.: (6*x^2/(1-x)^5)*hypergeometric2F0(3,5/2; - ; 6*x/(1-x)^2). - _G. C. Greubel_, Aug 16 2017

%t Join[{0, 0}, Table[4*n*(n - 1)*Pochhammer[1/2, n]*6^(n - 2)* Hypergeometric1F1[2 - n, -2*n, 2/3], {n, 2, 50}]] (* _G. C. Greubel_, Aug 14 2017 *)

%o (PARI) for(n=0,50, print1(sum(k=0,n-2, ((n+k+2)!/(4*k!*(n-k-2)!))*(3/2)^k), ", ")) \\ _G. C. Greubel_, Aug 14 2017

%Y Cf. A001516, A001518.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Dec 08 2001