A001514 Bessel polynomial {y_n}'(1). (Formerly M4654 N1993)

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

Offset

0,3

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

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)

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 *)

Prog

(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

Crossrefs

Cf. A001515, A001516, A001518, A065920, A144505.

Sequence in context: A199689 A181581 A137062 * A077364 A077486 A233021

Adjacent sequences: A001511 A001512 A001513 * A001515 A001516 A001517

Keyword

nonn

Author

N. J. A. Sloane

Status

approved