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A002801
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a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) with a(0) = a(1) = 1.
(Formerly M1882 N0744)
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6
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1, 1, 2, 8, 50, 418, 4348, 54016, 779804, 12824540, 236648024, 4841363104, 108748223128, 2660609220952, 70422722065040, 2005010410792832, 61098981903602192, 1984186236246187024, 68407835576255308576, 2495374564069015050880, 96019859122742736121376, 3886906732751071879958816, 165120572466718493379680192
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OFFSET
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0,3
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COMMENTS
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REFERENCES
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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
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FORMULA
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E.g.f.: exp(x/2)*(1-2*x)^(-1/4). - Paul Barry, Nov 26 2008
a(n) = hypergeom([1/4, -n],[],-4)/(2^n). - Mark van Hoeij, Jun 02 2010
a(n) ~ n^(n-1/4) * exp(-n+1/4) * Gamma(3/4) * 2^n / sqrt(Pi). - Vaclav Kotesovec, Oct 08 2013
0 = a(n)*(+a(n+1) - 3*a(n+2) + a(n+3)) + a(n+1)*(-a(n+1) + 3*a(n+2) - 2*a(n+3)) + a(n+2)*(+2*a(n+2)) if n>=0. - Michael Somos, Oct 30 2015
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EXAMPLE
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G.f. = 1 + x + 2*x^2 + 8*x^3 + 50*x^4 + 418*x^5 + 4348*x^6 + 54016*x^7 + 779804*x^8 + ...
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MATHEMATICA
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nxt[{n_, a_, b_}]:={n+1, b, b*(2n+1)-a*n}; Transpose[NestList[nxt, {1, 1, 1}, 30]][[2]] (* Harvey P. Dale, Sep 04 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[x/2] / (1 - 2 x)^(1/4), {x, 0, n}]]; (* Michael Somos, Oct 30 2015 *)
a[ n_] := If[ n < 0, 0, RecurrenceTable[{a[k] == (2 k - 1) a[k - 1] - (k - 1) a[k - 2], a[0] == a[1] == 1}, a, {k, n, n}]]; (* Michael Somos, Oct 30 2015 *)
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PROG
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(Maxima) a(n):=coeff(taylor(exp(x/2)/(1-2*x)^(1/4), x, 0, n), x, n)*n!;
(PARI) x='x+O('x^66); /* that many terms */
Vec(serlaplace(exp(x/2)*(1-2*x)^(-1/4))) /* show terms */ /* Joerg Arndt, Jul 10 2011 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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