A002801 a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) with a(0) = a(1) = 1. (Formerly M1882 N0744)

1, 1, 2, 8, 50, 418, 4348, 54016, 779804, 12824540, 236648024, 4841363104, 108748223128, 2660609220952, 70422722065040, 2005010410792832, 61098981903602192, 1984186236246187024, 68407835576255308576, 2495374564069015050880, 96019859122742736121376, 3886906732751071879958816, 165120572466718493379680192

Offset

0,3

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

Appears to be the BinomialMean transform of A007696 (see A075271). - John W. Layman, Oct 01 2002

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

Cf. A247249.

Sequence in context: A120956 A000557 A193352 * A322738 A233436 A225052

Adjacent sequences: A002798 A002799 A002800 * A002802 A002803 A002804

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from John W. Layman, Oct 01 2002

Status

approved