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A002119
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Bessel polynomial y_n(-2).
(Formerly M4444 N1880)
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9
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1, -1, 7, -71, 1001, -18089, 398959, -10391023, 312129649, -10622799089, 403978495031, -16977719590391, 781379079653017, -39085931702241241, 2111421691000680031, -122501544009741683039, 7597207150294985028449, -501538173463478753560673
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Absolute values give denominators of successive convergents to e using continued fraction 1+2/(1+1/(6+1/(10+1/(14+1/(18+1/(22+1/26...)))))).
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REFERENCES
| Leo Chao, Paul DesJarlais and John L Leonard, A binomial identity, via derangements, Math. Gaz. 89 (2005), 268-270..
L. Euler, 1737.
J. W. L. Glaisher, Reports of British Assoc. Adv. Sci., 1871, pp. 16-18.
D. H. Lehmer, Review of various tables by P. Pederson, Math. Comp., 2 (1946), 68-69.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
Index entries for sequences related to Bessel functions or polynomials
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FORMULA
| a(n) = -2(2n-1)*a(n-1) + a(n-2). - T. D. Noe, Oct 26 2006
If y = x + Sum_{k>1} A005363(k)*x^k/k!, then y = x + Sum{k>1} a(k-2)(-y)^k/k!. - Michael Somos Apr 02 2007
a(-n-1)= a(n). - Michael Somos Apr 02 2007
a(n)=(1/n!)*Int((-x(1+x))^n*exp(-(1+x)),x,-1,infinity). [From Paul Barry (pbarry(AT)wit.ie), Apr 19 2010]
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MATHEMATICA
| Table[(-1)^k (2k)! Hypergeometric1F1[-k, -2k, -1]/k!, {k, 0, 10}] (* Vladimir Reshetnikov (v.reshetnikov(AT)gmail.com), Feb 16 2011 *)
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PROG
| (PARI) {a(n)= if(n<0, n=-n-1); sum(k=0, n, (2*n-k)!/ (k!*(n-k)!)* (-1)^(n-k) )} /* Michael Somos Apr 02 2007 */
(PARI) {a(n)= local(A); if(n<0, n= -n-1); A= sqrt(1 +4*x +x*O(x^n)); n!*polcoeff( exp((A-1)/2)/A, n)} /* Michael Somos Apr 02 2007 */
(PARI) {a(n)= local(A); if(n<0, n= -n-1); n+=2 ; for(k= 1, n, A+= x*O(x^k); A= truncate( (1+x)* exp(A) -1-A) ); A+= x*O(x^n); A-= A^2; -(-1)^n*n!* polcoeff( serreverse(A), n)} /* Michael Somos Apr 02 2007 */
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CROSSREFS
| Cf. A001517, A053556, A053557, A001514, A065920, A065921, A065922, A065707, A000806, A006199, A065923.
See also A033815.
Polynomial coefficients are in A001498.
Sequence in context: A048552 A067307 A052390 * A146752 A022518 A113053
Adjacent sequences: A002116 A002117 A002118 * A002120 A002121 A002122
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KEYWORD
| sign,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 03 2000
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