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A089915
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Special values of generalized Laguerre polynomials L_n^(alpha)(x): a(n) = 4^n *n! *L_n^(-1/4)(-1).
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1
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1, 7, 93, 1747, 41881, 1214079, 41146933, 1592909707, 69246230193, 3336315914359, 176318707191181, 10135074699232707, 629237102421392713, 41946693027934462447, 2987346130479191856741, 226298375000985686615419, 18164688323228843027295073
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OFFSET
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0,2
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COMMENTS
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L_n^(alpha)(x) is orthogonal over the interval (0,infinity) with weight exp(-x)*x^alpha. L_1^(alpha)(x) = -x+1+alpha.
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LINKS
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FORMULA
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a(n) = (8*n-1)*a(n-1) - 4*(n-1)*(4*n-5)*a(n-2).
a(n) ~ 2^(2*n-1/2)*n^(n+1/8)*exp(2*sqrt(n)-n-1/2) * (1+97/(192*sqrt(n))).
(End)
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MATHEMATICA
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Table[4^n*n!*LaguerreL[n, -1/4, -1], {n, 0, 20}] (* Vaclav Kotesovec, Jul 31 2013 *)
RecurrenceTable[{a[n] == (8*n-1)*a[n-1] - 4*(n-1)*(4*n-5)*a[n-2], a[0] == 1, a[1] == 7}, a, {n, 0, 50}] (* G. C. Greubel, May 14 2018 *)
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PROG
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(Magma) I:=[7, 93]; [1] cat [n le 2 select I[n] else (8*n-1)*Self(n-1) - 4*(n-1)*(4*n-5)*Self(n-2): n in [1..30]]; // G. C. Greubel, May 14 2018
(PARI) m=30; v=concat([7, 93], vector(m-2)); for(n=3, m, v[n]=(8*n-1)*v[n-1]-4*(n-1)*(4*n-5)*v[n-2]); concat([1], v) \\ G. C. Greubel, May 14 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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