A002801 a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) with a(0) = a(1) = 1.
1, 1, 2, 8, 50, 418, 4348, 54016, 779804, 12824540, 236648024, 4841363104, 108748223128, 2660609220952, 70422722065040, 2005010410792832, 61098981903602192, 1984186236246187024, 68407835576255308576, 2495374564069015050880, 96019859122742736121376, 3886906732751071879958816, 165120572466718493379680192
Offset: 0
Keywords
Examples
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 + ...
References
- 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
- Vincenzo Librandi, Table of n, a(n) for n = 0..100
- E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 223.
- E. Lucas, Theorie des nombres (annotated scans of a few selected pages)
- Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
- J. J. Sylvester, Note on determinants and duadic disynthemes, American J of Math, Vol 2 No 1, (1879), 89-96, circa p. 94.
Crossrefs
Cf. A247249.
Programs
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Mathematica
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_] := HypergeometricPFQ[{1/4, -n}, {}, -4]/(2^n); Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Mar 17 2014, after Mark van Hoeij *) 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 *) -
Maxima
a(n):=coeff(taylor(exp(x/2)/(1-2*x)^(1/4),x,0,n),x,n)*n!; makelist(a(n),n,0,12); /* Emanuele Munarini, Jul 07 2011 */
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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 */
Formula
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
Extensions
More terms from John W. Layman, Oct 01 2002
Comments