OFFSET
0,3
COMMENTS
Row sums of A152148. - Paul Barry, Nov 26 2008
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.
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
EXAMPLE
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 + ...
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 *)
PROG
(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 */
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from John W. Layman, Oct 01 2002
STATUS
approved