A105523 Expansion of 1-x*c(-x^2) where c(x) is the g.f. of A000108.

1, -1, 0, 1, 0, -2, 0, 5, 0, -14, 0, 42, 0, -132, 0, 429, 0, -1430, 0, 4862, 0, -16796, 0, 58786, 0, -208012, 0, 742900, 0, -2674440, 0, 9694845, 0, -35357670, 0, 129644790, 0, -477638700, 0, 1767263190, 0

Offset

0,6

Comments

Row sums of A105522. Row sums of inverse of A105438.

First column of number triangle A106180.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 313.

R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.

Formula

G.f.: (1 + 2*x - sqrt(1+4*x^2))/(2*x).

a(n) = 0^n + sin(Pi*(n-2)/2)(C((n-1)/2)(1-(-1)^n)/2).

G.f.: 1/(1+x/(1-x/(1+x/(1-x/(1+x/(1-x.... (continued fraction). - Paul Barry, Jan 15 2009

a(n) = Sum{k = 0..n} A090181(n,k)*(-1)^k. - Philippe Deléham, Feb 02 2009

a(n) = (1/n)*sum_{i = 0..n-1} (-2)^i*binomial(n, i)*binomial(2*n-i-2, n-1). - Vladimir Kruchinin, Dec 26 2010

With offset 1, a(n) = -2 * a(n-1) + Sum_{k=1..n-1} a(k) * a(n-k), for n>1. - Michael Somos, Jul 25 2011

D-finite with recurrence: (n+3)*a(n+2) = -4*n*a(n), a(0)=1, a(1)=-1. - Fung Lam, Mar 18 2014

For nonzero terms, a(n) ~ (-1)^((n+1)/2)/sqrt(2*Pi)*2^(n+1)/(n+1)^(3/2). - Fung Lam, Mar 17 2014

a(n) = -(sqrt(Pi)*2^(n-1))/(Gamma(1-n/2)*Gamma((n+3)/2)) for n odd. - Peter Luschny, Oct 31 2014

From Peter Bala, Apr 20 2024: (Start)

a(n) = Sum_{k = 0..n} (-2)^(n-k)*binomial(n + k, 2*k)*Catalan(k), where Catalan(k) = A000108(k).

a(n) = (-2)^n * hypergeom([-n, n+1], [2], 1/2).

O.g.f.: A(x) = 1/x * series reversion of x*(1 - x)/(1 - 2*x). Cf. A152681. (End)

Example

G.f. = 1 - x + x^3 - 2*x^5 + 5*x^7 - 14*x^9 + 42*x^11 - 132*x^13 + 429*x^15 + ...

Maple

A105523_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w]:=-a[w-1]+(-1)^w*add(a[j]*a[w-j-1], j=1..w-1) od; convert(a, list)end: A105523_list(40); # Peter Luschny, May 19 2011

Mathematica

a[n_?EvenQ] := 0; a[n_?OddQ] := 4^n*Gamma[n/2] / (Gamma[-n/2]*(n+1)!); a[0] = 1; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 14 2011, after Vladimir Kruchinin *)
CoefficientList[Series[(1 + 2 x - Sqrt[1 + 4 x^2])/(2 x), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 01 2014 *)
a[ n_] := SeriesCoefficient[ (1 + 2 x - Sqrt[ 1 + 4 x^2]) / (2 x), {x, 0, n}]; (* Michael Somos, Jun 17 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], a[n] = -2 a[n - 1] + Sum[ a[j] a[n - j - 1], {j, 0, n - 1}]]; (* Michael Somos, Jun 17 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, n++; A = vector(n); A[1] = 1; for( k=2, n, A[k] = -2 * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 24 2011 */
(Sage)
def A105523(n):
    if is_even(n): return 0 if n>0 else 1
    return -(sqrt(pi)*2^(n-1))/(gamma(1-n/2)*gamma((n+3)/2))
[A105523(n) for n in (0..29)] # Peter Luschny, Oct 31 2014
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 + 2*x - Sqrt(1+4*x^2))/(2*x))); // G. C. Greubel, Sep 16 2018

Crossrefs

Cf. A000108, A097331, A090192, A090181, A105522, A105438.

Sequence in context: A361921 A245928 A242839 * A210628 A090192 A097331

Adjacent sequences: A105520 A105521 A105522 * A105524 A105525 A105526

Keyword

easy,sign

Author

Paul Barry, Apr 11 2005

Extensions

Typo in definition corrected by Robert Israel, Oct 31 2014

Status

approved