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A118351 Central terms of pendular triangle A118350. 5
1, 1, 6, 42, 325, 2688, 23286, 208659, 1918314, 17994264, 171542460, 1657212768, 16188521454, 159634359415, 1586932321578, 15886925400954, 160026976985205, 1620715748715648, 16493797802077032, 168583560794745684 (list; graph; refs; listen; history; text; internal format)
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 A218755 A165314

Adjacent sequences:  A118348 A118349 A118350 * A118352 A118353 A118354

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Apr 26 2006

STATUS

approved

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Last modified May 6 19:52 EDT 2021. Contains 343586 sequences. (Running on oeis4.)