A118351 - OEIS

%I #25 Feb 19 2021 03:53:16

%S 1,1,6,42,325,2688,23286,208659,1918314,17994264,171542460,1657212768,

%T 16188521454,159634359415,1586932321578,15886925400954,

%U 160026976985205,1620715748715648,16493797802077032,168583560794745684

%N Central terms of pendular triangle A118350.

%H G. C. Greubel, <a href="/A118351/b118351.txt">Table of n, a(n) for n = 0..500</a>

%F G.f. A=A(x) satisfies: A = 1 - 3*x*A + 3*x*A^2 + x*A^3.

%F G.f.: 1 + Series_Reversion( x/((1+x)*(1+5*x+x^2)) ).

%F G.f.: (1/x)*Series_Reversion( x*(1-3*x+sqrt((1-3*x)*(1-7*x)))/2/(1-3*x) ).

%F 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 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 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 Table[T[2*n,n,3], {n,0,30}] (* _G. C. Greubel_, Feb 18 2021 *)

%t 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 (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 for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n)=polcoeff(1 + serreverse( x/((1+x)*(1+5*x+x^2 +x*O(x^n)))),n)}

%o for(n=0,30,print1(a(n),", "))

%o (Sage)

%o def S_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( (x/((1+x)*(1+5*x+x^2))).reverse() ).list()

%o a=S_list(31); [1]+a[1:] # _G. C. Greubel_, Feb 18 2021

%o (Magma)

%o R<x>:=PowerSeriesRing(Rationals(), 30);

%o [1] cat Coefficients(R!( Reversion( x/((1+x)*(1+5*x+x^2)) ) )); // _G. C. Greubel_, Feb 18 2021

%Y Cf. A118350, A118352, A118353, A118354.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 26 2006