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A001516 Bessel polynomial {y_n}''(1).
(Formerly M4295 N1795)
12
0, 0, 6, 120, 1980, 32970, 584430, 11204676, 233098740, 5254404210, 127921380840, 3350718545460, 94062457204716, 2819367702529560, 89912640142178490, 3040986592542420060, 108752084073199561140, 4101112025363285051526 (list; graph; refs; listen; history; text; internal format)
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

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

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

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A170917 A115678 A048604 * A350712 A026337 A223629

Adjacent sequences: A001513 A001514 A001515 * A001517 A001518 A001519

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 30 04:37 EST 2022. Contains 358431 sequences. (Running on oeis4.)