A001516 Bessel polynomial {y_n}''(1).

0, 0, 6, 120, 1980, 32970, 584430, 11204676, 233098740, 5254404210, 127921380840, 3350718545460, 94062457204716, 2819367702529560, 89912640142178490, 3040986592542420060, 108752084073199561140, 4101112025363285051526

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

• G. C. Greubel, Table of n, a(n) for n = 0..400
• 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

Crossrefs

Cf. A001497, A001498, A001514, A001515, A001518, A065944, A144505.

Programs

Maple

(As in A001497 define:) f := proc(n) option remember; if n <=1 then (1+x)^n else expand((2*n-1)*x*f(n-1)+f(n-2)); fi; end;
[seq( subs(x=1,diff(f(n),x$2)),n=0..60)];

Mathematica

Table[Sum[(n+k+2)!/(2^(k+2)*(n-k-2)!*k!), {k,0,n-2}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)
Join[{0, 0}, Table[n*(n - 1)*Pochhammer[1/2, n]*2^n* Hypergeometric1F1[2 - n, -2*n, 2], {n,2,50}]] (* G. C. Greubel, Aug 14 2017 *)

PARI

for(n=0,20, print1(sum(k=0,n-2, (n+k+2)!/(2^(k+2)*(n-k-2)!*k!)), ", ")) \\ G. C. Greubel, Aug 14 2017

Formula

G.f.: 6*x^2*(1-x)^(-5)*hypergeom([5/2,3],[],2*x/(x-1)^2). - Mark van Hoeij, Nov 07 2011

D-finite with recurrence: (n-2)*(n-1)*a(n) = (2*n - 1)*(n^2 - n + 2)*a(n-1) + n*(n+1)*a(n-2). - Vaclav Kotesovec, Jul 22 2015

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

a(n) = n*(n - 1)*(1/2){n}*2^n* hypergeometric1F1(2 - n, -2*n, 2), where (a){n} is the Pochhammer symbol. - G. C. Greubel, Aug 14 2017

E.g.f.: (-1)*(1 - 2*x)^(-5/2)*((4 - 14*x + 9*x^2)*sqrt(1 - 2*x) + (2*x^3 - 24*x^2 + 18*x - 4))*exp((1 - sqrt(1 - 2*x))). - G. C. Greubel, Aug 16 2017