OFFSET
0,2
COMMENTS
2*a(n-1) is the second diagonal of the triangle A115195.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
G.f. is the series reversion of -((1 + 5*x + 2*x^2 - (1 + 2*x)*sqrt(1 + 6*x + x^2))/(4*x^2)).
G.f.: 2*((c(2*x))^3)/(1+c(2*x)) with the o.g.f. c(x) of A000108 (Catalan numbers).
a(n) = Sum_{j=1..n+1} A115195(n, j), n >= 0.
G.f.: (-1 + (1-x)*c(2*x))/(x*(1+x)); cf. A115139. - Wolfdieter Lang, Feb 23 2006
D-finite with recurrence: (n+2)*(7*n-5)*a(n) = (7*n-2)*(7*n-1)*a(n-1) + 4*(2*n-1)*(7*n+2)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ 7*2^(3n+3)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
D-finite with recurrence (n+2)*a(n) = 2*(4*n+1)*a(n-1) + (n+16)*a(n-2) - 4*(2*n-3)*a(n-3). - R. J. Mathar, Mar 10 2022
a(n) = ( (7*n-1)*(7*n-2)*a(n-1) + 4*(2*n-1)*(7*n+2)*a(n-2) )/((n+2)*(7*n-5)), with a(0) = 1, a(1) = 5. - G. C. Greubel, Nov 23 2022
EXAMPLE
{1}
{-2,-1,0,1,2}
{-3,-2,-1,0,1,2,3,-2,-1,0,1,2,-1,0,1,-2,-1,0,1,2,-3,-2,-1,0,1,2,3}
MATHEMATICA
Rest@CoefficientList[InverseSeries[Series[ -((1+5*n+2*n^2-(1+2*n)*Sqrt[1+6*n+n^2] )/(4*n^2)), {n, 0, 28}]], n] or Length/@Flatten/@NestList[ # /. k_Integer :> Range[ -1-Abs[k], Abs[k]+1]&, {1}, 8]
Flatten[{1, RecurrenceTable[{(n+2)*(7*n-5)*a[n] == (7*n-2)*(7*n-1)*a[n-1] + 4*(2*n-1)*(7*n+2)*a[n-2], a[1]==5, a[2]==27}, a, {n, 20}]}] (* Vaclav Kotesovec, Oct 14 2012 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-5*x -(1- x)*Sqrt(1-8*x))/(4*x^2*(1+x)) )); // G. C. Greubel, Nov 23 2022
(Sage)
def A084076_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1-5*x -(1-x)*sqrt(1-8*x))/(4*x^2*(1+x)) ).list()
A084076_list(40) # G. C. Greubel, Nov 23 2022
(PARI) {a(n) = my(L); L = [1]; if(n < 0, 0, for(i = 1, n, L = concat([ vector(3 + 2*abs(k), i, i - abs(k) - 2) | k <- L])); #L)}; /* Michael Somos, Nov 23 2022 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Wouter Meeussen, May 11 2003
STATUS
approved