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%I M1882 N0744 #50 Oct 30 2015 11:42:38
%S 1,1,2,8,50,418,4348,54016,779804,12824540,236648024,4841363104,
%T 108748223128,2660609220952,70422722065040,2005010410792832,
%U 61098981903602192,1984186236246187024,68407835576255308576,2495374564069015050880,96019859122742736121376,3886906732751071879958816,165120572466718493379680192
%N a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) with a(0) = a(1) = 1.
%C Row sums of A152148. - _Paul Barry_, Nov 26 2008
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A002801/b002801.txt">Table of n, a(n) for n = 0..100</a>
%H E. Lucas, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k29021h">Théorie des Nombres</a>, Gauthier-Villars, Paris, 1891, Vol. 1, p. 223.
%H E. Lucas, <a href="/A000899/a000899.pdf">Theorie des nombres</a> (annotated scans of a few selected pages)
%H Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%H J. J. Sylvester, <a href="http://www.jstor.org/stable/2369200">Note on determinants and duadic disynthemes</a>, American J of Math, Vol 2 No 1, (1879), 89-96, circa p. 94.
%F Appears to be the BinomialMean transform of A007696 (see A075271). - _John W. Layman_, Oct 01 2002
%F E.g.f.: exp(x/2)*(1-2*x)^(-1/4). - _Paul Barry_, Nov 26 2008
%F a(n) = hypergeom([1/4, -n],[],-4)/(2^n). - _Mark van Hoeij_, Jun 02 2010
%F a(n) ~ n^(n-1/4) * exp(-n+1/4) * Gamma(3/4) * 2^n / sqrt(Pi). - _Vaclav Kotesovec_, Oct 08 2013
%F 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
%e 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 + ...
%t 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 *)
%t a[n_] := HypergeometricPFQ[{1/4, -n}, {}, -4]/(2^n); Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Mar 17 2014, after _Mark van Hoeij_ *)
%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[x/2] / (1 - 2 x)^(1/4), {x, 0, n}]]; (* _Michael Somos_, Oct 30 2015 *)
%t 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 *)
%o (Maxima) a(n):=coeff(taylor(exp(x/2)/(1-2*x)^(1/4),x,0,n),x,n)*n!;
%o makelist(a(n),n,0,12); /* _Emanuele Munarini_, Jul 07 2011 */
%o (PARI) x='x+O('x^66); /* that many terms */
%o Vec(serlaplace(exp(x/2)*(1-2*x)^(-1/4))) /* show terms */ /* _Joerg Arndt_, Jul 10 2011 */
%Y Cf. A247249.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _John W. Layman_, Oct 01 2002
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