A369123
Expansion of (1/x) * Series_Reversion( x * ((1-x)^2+x^2) ).
Original entry on oeis.org
1, 2, 6, 20, 68, 224, 672, 1584, 880, -22880, -215072, -1414400, -8012032, -41344000, -198120448, -884348160, -3640426752, -13403384320, -40424947200, -65476561920, 329862128640, 4603911045120, 35276325027840, 221747978649600, 1244854463643648
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^2+x^2))/x)
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a(n) = sum(k=0, n\2, (-1)^k*binomial(n+k, k)*binomial(3*n+1, n-2*k))/(n+1);
A059491
Expansion of generating function A_{QT}^(1)(4n;3).
Original entry on oeis.org
1, 1, 6, 189, 30618, 25332021, 106698472452, 2283997201168644, 248218139523497121576, 136861610819571430116630660, 382684747771430768732371981946100, 5424628155237728987530088501811168904125, 389729317367139375014273384868937660572301897500
Offset: 0
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f[n_] := Product[(3 k + 1)!/(n + k)!, {k, 0, n - 1}]; Table[3^(n*(n - 1)/2)*f[n], {n,0,20}] (* G. C. Greubel, Sep 10 2017 *)
A097187
Antidiagonal sums of triangle A097186, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A057083(y)^(n+1), where R_n(1/3) = 3^n for all n>=0.
Original entry on oeis.org
1, 1, 7, 10, 58, 94, 499, 868, 4360, 7951, 38407, 72508, 339997, 659380, 3019639, 5984968, 26880052, 54249628, 239683171, 491235070, 2139947788, 4444675456, 19125212575, 40190140696, 171064560433, 363227946394, 1531088393647
Offset: 0
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R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 3*x/((1-9*x^2) + (3*x-1)*(1-9*x^2)^(2/3)) )); // G. C. Greubel, Sep 17 2019
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seq(coeff(series(3*x/((1-9*x^2) +(3*x-1)*(1-9*x^2)^(2/3)), x, n+2), x, n), n = 0..30); # G. C. Greubel, Sep 17 2019
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CoefficientList[Series[3*x/((1-9*x^2) +(3*x-1)*(1-9*x^2)^(2/3)), {x, 0, 30}], x] (* G. C. Greubel, Sep 17 2019 *)
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a(n)=polcoeff(3*x/((1-9*x^2)+(3*x-1)*(1-9*x^2+x^2*O(x^n))^(2/3)), n,x)
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def A097187_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P(3*x/((1-9*x^2) + (3*x-1)*(1-9*x^2)^(2/3))).list()
A097187_list(30) # G. C. Greubel, Sep 17 2019