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A118351 Central terms of pendular triangle A118350.

%I A118351 #25 Feb 19 2021 03:53:16
%S A118351 1,1,6,42,325,2688,23286,208659,1918314,17994264,171542460,1657212768,
%T A118351 16188521454,159634359415,1586932321578,15886925400954,
%U A118351 160026976985205,1620715748715648,16493797802077032,168583560794745684
%N A118351 Central terms of pendular triangle A118350.
%H A118351 G. C. Greubel, <a href="/A118351/b118351.txt">Table of n, a(n) for n = 0..500</a>
%F A118351 G.f. A=A(x) satisfies: A = 1 - 3*x*A + 3*x*A^2 + x*A^3.
%F A118351 G.f.: 1 + Series_Reversion( x/((1+x)*(1+5*x+x^2)) ).
%F A118351 G.f.: (1/x)*Series_Reversion( x*(1-3*x+sqrt((1-3*x)*(1-7*x)))/2/(1-3*x) ).
%F A118351 For n>0: a(n) = 1/n*sum(j=0..n, C(n,j) *sum(i=0..(n-1), C(j,i)*C(n-j,2*j-n-i-1) *6^(2*n-3*j+2*i+1))). - _Vladimir Kruchinin_, Dec 26 2010
%F A118351 a(n) ~ s^(3/2) / (3*sqrt(2*Pi*(1 + 3*s + 3*s^2)) * n^(3/2) * r^(n+1)), where s = 2*sin(Pi/6 + arctan(sqrt(7)/3)/3) - 1, r = 2*s/(9 - 12*sin(Pi/6 - 2*arctan(sqrt(7)/3)/3)). - _Vaclav Kotesovec_, Feb 18 2021
%t A118351 T[n_, k_, p_]:= T[n,k,p] = If[n<k || k<0, 0, If[k==0, 1, If[k==n, 0, If[n<=2*k, T[n,n-k-1,p] + p*T[n-1,k,p], T[n,n-k,p] + T[n-1,k,p] ]]]];
%t A118351 Table[T[2*n,n,3], {n,0,30}] (* _G. C. Greubel_, Feb 18 2021 *)
%t A118351 Join[{1}, Rest@CoefficientList[InverseSeries[Series[ x/((1+x)*(1+5*x+x^2)), {x,0,30}]], x]] (* _G. C. Greubel_, Feb 18 2021 *)
%o A118351 (PARI) {a(n)=polcoeff((serreverse(x*(1-3*x+sqrt((1-3*x)*(1-7*x)+x*O(x^n)))/2/(1-3*x))/x),n)}
%o A118351 for(n=0,30,print1(a(n),", "))
%o A118351 (PARI) {a(n)=polcoeff(1 + serreverse( x/((1+x)*(1+5*x+x^2 +x*O(x^n)))),n)}
%o A118351 for(n=0,30,print1(a(n),", "))
%o A118351 (Sage)
%o A118351 def S_list(prec):
%o A118351     P.<x> = PowerSeriesRing(ZZ, prec)
%o A118351     return P( (x/((1+x)*(1+5*x+x^2))).reverse() ).list()
%o A118351 a=S_list(31); [1]+a[1:] # _G. C. Greubel_, Feb 18 2021
%o A118351 (Magma)
%o A118351 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A118351 [1] cat Coefficients(R!( Reversion( x/((1+x)*(1+5*x+x^2)) ) )); // _G. C. Greubel_, Feb 18 2021
%Y A118351 Cf. A118350, A118352, A118353, A118354.
%K A118351 nonn
%O A118351 0,3
%A A118351 _Paul D. Hanna_, Apr 26 2006