A001514 Bessel polynomial {y_n}'(1).

0, 1, 9, 81, 835, 9990, 137466, 2148139, 37662381, 733015845, 15693217705, 366695853876, 9289111077324, 253623142901401, 7425873460633005, 232122372003909045, 7715943399320562331, 271796943164015920914, 10114041937573463433966

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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
• N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
• Index entries for sequences related to Bessel functions or polynomials

Crossrefs

Cf. A001515, A001516, A001518, A065920, 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)),n=0..60)];
f2:=proc(n) local k; add((n+k+2)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f2(n),n=0..60)]; # uses a different offset

Mathematica

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

PARI

for(n=0,50, print1(sum(k=0,n-1, (n+k+1)!/((n-k-1)!*k!*2^(k+1))), ", ")) \\ G. C. Greubel, Aug 14 2017

Formula

a(n) = (1/2) * Sum_{k=0..n} (n+k+2)!/((n-k)!*k!*2^k) (with a different offset).

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

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

a(n) = n*2^n*(1/2){n}*hypergeometric1f1(1-n, -2*n, 2), where (a){n} is the Pochhammer symbol. - G. C. Greubel, Aug 14 2017

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

G.f.: (1/(1-t))*hypergeometric2f0(2, 3/2; -; 2*t/(1-t)^2).

E.g.f.: (1 - 2*x)^(-3/2)*((1 - x)*sqrt(1 - 2*x) + (3*x - 1))*exp((1 - sqrt(1 - 2*x))). (End)