A097190
Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/3) = 9^n, where R_n(y) forms the initial (n+1) terms of g.f. A097191(y)^(n+1).
Original entry on oeis.org
1, 1, 24, 1, 36, 612, 1, 48, 1104, 15912, 1, 60, 1740, 32130, 417690, 1, 72, 2520, 56700, 912492, 11027016, 1, 84, 3444, 91350, 1750014, 25562628, 292215924, 1, 96, 4512, 137808, 3059856, 52303968, 710025264, 7764594552, 1, 108, 5724, 197802, 4992354
Offset: 0
Row polynomials evaluated at y=1/3 equals powers of 9:
9^1 = 1 + 24/3;
9^2 = 1 + 36/3 + 612/3^2;
9^3 = 1 + 48/3 + 1104/3^2 + 15912/3^3;
9^4 = 1 + 60/3 + 1740/3^2 + 32130/3^3 + 417690/3^4;
where A097191(y)^(n+1) has the same initial terms as the n-th row:
A097191(y) = 1 + 12y + 60y^2 + 90y^3 - 558y^4 - 2916y^5 + 2160y^6 +...
A097191(y)^2 = 1 + 24y +...
A097191(y)^3 = 1 + 36y + 612y^2 +...
A097191(y)^4 = 1 + 48y + 1104y^2 + 15912y^3 +...
A097191(y)^5 = 1 + 60y + 1740y^2 + 32130y^3 + 417690y^4 +...
Rows begin with n=0:
1;
1, 24;
1, 36, 612;
1, 48, 1104, 15912;
1, 60, 1740, 32130, 417690;
1, 72, 2520, 56700, 912492, 11027016;
1, 84, 3444, 91350, 1750014, 25562628, 292215924;
1, 96, 4512, 137808, 3059856, 52303968, 710025264, 7764594552; ...
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Table[SeriesCoefficient[3*y/((1-27*x*y) + (3*y-1)*(1-27*x*y)^(8/9)), {x, 0,n}, {y,0,k}], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 17 2019 *)
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{T(n,k)=if(n==0,1,if(k==0,1,if(k==n, 3^n*(9^n-sum(j=0,n-1, T(n,j)/3^j)), polcoeff((Ser(vector(n,i,T(n-1,i-1)),x) +x*O(x^k))^((n+1)/n),k,x))))}
A097193
G.f. A(x) satisfies A097191(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097190.
Original entry on oeis.org
1, 12, 204, 3978, 83538, 1837836, 41745132, 970574319, 22970258883, 551286213192, 13381219902024, 327839887599588, 8095123378420596, 201221638263597672, 5030540956589941800, 126392341534322287725
Offset: 0
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R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (1-(1-27*x)^(1/9))/(3*x) )); // G. C. Greubel, Sep 17 2019
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seq(coeff(series((1-(1-27*x)^(1/9))/(3*x), x, n+2), x, n), n = 0 ..20); # G. C. Greubel, Sep 17 2019
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CoefficientList[Series[(1-(1-27*x)^(1/9))/(3*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
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a(n)=polcoeff((1-(1-27*x+x^2*O(x^n))^(1/9))/(3*x),n,x)
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def A097193_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P((1-(1-27*x)^(1/9))/(3*x)).list()
A097193_list(20) # G. C. Greubel, Sep 17 2019
A097192
Main diagonal of triangle A097190, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A097191(y)^(n+1), where R_n(1/3) = 9^n for all n>=0.
Original entry on oeis.org
1, 24, 612, 15912, 417690, 11027016, 292215924, 7764594552, 206732329947, 5512862131920, 147193418922264, 3934078651195056, 105236603919467748, 2817102935690367408, 75458114348849127000, 2022277464549156603600
Offset: 0
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R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 1/(1-27*x)^(8/9) )); // G. C. Greubel, Sep 17 2019
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seq(coeff(series(1/(1-27*x)^(8/9), x, n+1), x, n), n = 0 ..20); # G. C. Greubel, Sep 17 2019
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CoefficientList[Series[(1-27*x)^(-8/9), {x,0,20}], x] (* G. C. Greubel, Sep 17 2019 *)
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a(n)=polcoeff(1/(1-27*x+x*O(x^n))^(8/9),n,x)
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def A097192_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P(1/(1-27*x)^(8/9)).list()
A097192_list(20) # G. C. Greubel, Sep 17 2019
A097194
Row sums of triangle A097190, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A097191(y)^(n+1), where R_n(1/3) = 9^n for all n>=0.
Original entry on oeis.org
1, 25, 649, 17065, 451621, 11998801, 319623445, 8530126057, 227974775239, 6099550226965, 163340461497907, 4377292845062689, 117376545230379631, 3149059523347103293, 84522568856319875179, 2269506752111508954553
Offset: 0
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R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 3/((1-27*x) +2*(1-27*x)^(8/9)) )); // G. C. Greubel, Sep 17 2019
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seq(coeff(series(3/((1-27*x) +2*(1-27*x)^(8/9)), x, n+1), x, n), n = 0 ..20); # G. C. Greubel, Sep 17 2019
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CoefficientList[Series[3/((1-27*x) +2*(1-27*x)^(8/9)), {x,0,20}], x] (* G. C. Greubel, Sep 17 2019 *)
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a(n)=polcoeff(3/((1-27*x) + 2*(1-27*x+x*O(x^n))^(8/9)),n,x)
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def A097194_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P(3/((1-27*x) +2*(1-27*x)^(8/9))).list()
A097194_list(20) # G. C. Greubel, Sep 17 2019
Showing 1-4 of 4 results.