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

%I A084076 #35 Nov 25 2022 13:16:32
%S A084076 1,5,27,157,963,6141,40323,270845,1852419,12857341,90337283,641286141,
%T A084076 4592533507,33139654653,240723001347,1758796578813,12916805074947,
%U A084076 95300512382973,706044251602947,5250379998560253,39176121681444867
%N A084076 Length of list created by n substitutions k -> Range[-1-abs(k), abs(k)+1] starting with {1}.
%C A084076 2*a(n-1) is the second diagonal of the triangle A115195.
%C A084076 Row sums of A167432. Hankel transform is A167435. - _Paul Barry_, Nov 03 2009
%H A084076 Vincenzo Librandi, <a href="/A084076/b084076.txt">Table of n, a(n) for n = 0..200</a>
%F A084076 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)).
%F A084076 G.f.: 2*((c(2*x))^3)/(1+c(2*x)) with the o.g.f. c(x) of A000108 (Catalan numbers).
%F A084076 a(n) = Sum_{j=1..n+1} A115195(n, j), n >= 0.
%F A084076 G.f.: (-1 + (1-x)*c(2*x))/(x*(1+x)); cf. A115139. - _Wolfdieter Lang_, Feb 23 2006
%F A084076 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
%F A084076 a(n) ~ 7*2^(3n+3)/(9*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 14 2012
%F A084076 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
%F A084076 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
%e A084076 {1}
%e A084076 {-2,-1,0,1,2}
%e A084076 {-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}
%t A084076 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]
%t A084076 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 *)
%o A084076 (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
%o A084076 (Sage)
%o A084076 def A084076_list(prec):
%o A084076     P.<x> = PowerSeriesRing(ZZ, prec)
%o A084076     return P( (1-5*x -(1-x)*sqrt(1-8*x))/(4*x^2*(1+x)) ).list()
%o A084076 A084076_list(40) # _G. C. Greubel_, Nov 23 2022
%o A084076 (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 */
%Y A084076 Third column (m=2) of triangle A115193, called C(1, 2).
%Y A084076 Cf. A000108, A115139, A115195, A167432, A167435.
%Y A084076 Cf. A084075, A084077, A084078.
%K A084076 nonn
%O A084076 0,2
%A A084076 _Wouter Meeussen_, May 11 2003