A084076 Length of list created by n substitutions k -> Range[-1-abs(k), abs(k)+1] starting with {1}.

1, 5, 27, 157, 963, 6141, 40323, 270845, 1852419, 12857341, 90337283, 641286141, 4592533507, 33139654653, 240723001347, 1758796578813, 12916805074947, 95300512382973, 706044251602947, 5250379998560253, 39176121681444867

Offset

0,2

Comments

2*a(n-1) is the second diagonal of the triangle A115195.

Row sums of A167432. Hankel transform is A167435. - Paul Barry, Nov 03 2009

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

Third column (m=2) of triangle A115193, called C(1, 2).

Cf. A000108, A115139, A115195, A167432, A167435.

Cf. A084075, A084077, A084078.

Sequence in context: A357227 A101386 A153233 * A355252 A337011 A081924

Adjacent sequences: A084073 A084074 A084075 * A084077 A084078 A084079

Keyword

nonn

Author

Wouter Meeussen, May 11 2003

Status

approved