A118351 Central terms of pendular triangle A118350.

1, 1, 6, 42, 325, 2688, 23286, 208659, 1918314, 17994264, 171542460, 1657212768, 16188521454, 159634359415, 1586932321578, 15886925400954, 160026976985205, 1620715748715648, 16493797802077032, 168583560794745684

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

Formula

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

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

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

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

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

Mathematica

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] ]]]];
Table[T[2*n, n, 3], {n, 0, 30}] (* G. C. Greubel, Feb 18 2021 *)
Join[{1}, Rest@CoefficientList[InverseSeries[Series[ x/((1+x)*(1+5*x+x^2)), {x, 0, 30}]], x]] (* G. C. Greubel, Feb 18 2021 *)

Prog

(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)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(1 + serreverse( x/((1+x)*(1+5*x+x^2 +x*O(x^n)))), n)}
for(n=0, 30, print1(a(n), ", "))
(Sage)
def S_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (x/((1+x)*(1+5*x+x^2))).reverse() ).list()
a=S_list(31); [1]+a[1:] # G. C. Greubel, Feb 18 2021
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
[1] cat Coefficients(R!( Reversion( x/((1+x)*(1+5*x+x^2)) ) )); // G. C. Greubel, Feb 18 2021

Crossrefs

Cf. A118350, A118352, A118353, A118354.

Sequence in context: A145301 A107266 A142985 * A033296 A364437 A218755

Adjacent sequences: A118348 A118349 A118350 * A118352 A118353 A118354

Keyword

nonn

Author

Paul D. Hanna, Apr 26 2006

Status

approved