A097181
Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 8^n, where R_n(y) forms the initial (n+1) terms of g.f. A097182(y)^(n+1).
Original entry on oeis.org
1, 1, 14, 1, 21, 210, 1, 28, 378, 3220, 1, 35, 595, 6475, 49910, 1, 42, 861, 11396, 108402, 778596, 1, 49, 1176, 18326, 207074, 1791930, 12198004, 1, 56, 1540, 27608, 361018, 3647672, 29389492, 191682920, 1, 63, 1953, 39585, 587727, 6783147
Offset: 0
Row polynomials evaluated at y=1/2 equals powers of 8:
8^1 = 1 + 14/2;
8^2 = 1 + 21/2 + 210/2^2;
8^3 = 1 + 28/2 + 378/2^2 + 3220/2^3;
8^4 = 1 + 35/2 + 595/2^2 + 6475/2^3 + 49910/2^4;
where A097182(y)^(n+1) has the same initial terms as the n-th row:
A097182(y) = 1 + 7*x + 21*x^2 + 21*x^3 - 63*x^4 - 231*x^5 -+...
A097182(y)^2 = 1 + 14y +...
A097182(y)^3 = 1 + 21y + 210y^2 +...
A097182(y)^4 = 1 + 28y + 378y^2 + 3220y^3 +...
A097182(y)^5 = 1 + 35y + 595y^2 + 6475y^3 + 49910y^4 +...
Rows begin with n=0:
1;
1, 14;
1, 21, 210;
1, 28, 378, 3220;
1, 35, 595, 6475, 49910;
1, 42, 861, 11396, 108402, 778596;
1, 49, 1176, 18326, 207074, 1791930, 12198004;
1, 56, 1540, 27608, 361018, 3647672, 29389492, 191682920;
1, 63, 1953, 39585, 587727, 6783147, 62974371, 479497491, 3019005990; ...
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Table[SeriesCoefficient[2*y/((1-16*x*y) + (2*y-1)*(1-16*x*y)^(7/8)), {x, 0,n}, {y,0,k}], {n,0,10}, {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, 2^n*(4^n -sum(j=0,n-1, T(n,j)/2^j)), polcoeff((Ser(vector(n,i,T(n-1,i-1)),x) +x*O(x^k))^((n+1)/n),k,x))))}
A097184
G.f. A(x) satisfies A097182(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097181.
Original entry on oeis.org
1, 7, 70, 805, 9982, 129766, 1742572, 23960365, 335445110, 4763320562, 68418604436, 992069764322, 14499481170860, 213349508656940, 3157572728122712, 46968894330825341, 701770538825272742, 10526558082379091130, 158452400608443161220
Offset: 0
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R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (1-(1-16*x)^(1/8))/(2*x) )); // G. C. Greubel, Sep 17 2019
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seq(coeff(series((1-(1-16*x)^(1/8))/(2*x), x, n+2), x, n), n = 0..20); # G. C. Greubel, Sep 17 2019
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CoefficientList[Series[(1-(1-16*x)^(1/8))/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 09 2014 *)
Table[FullSimplify[16^n*Gamma[n+7/8]/(Gamma[7/8]*Gamma[n+2])], {n, 0, 20}] (* Vaclav Kotesovec, Feb 09 2014 *)
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a(n)=polcoeff((1-(1-16*x+x^2*O(x^n))^(1/8))/(2*x),n,x)
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def A097184_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P((1-(1-16*x)^(1/8))/(2*x)).list()
A097184_list(20) # G. C. Greubel, Sep 17 2019
A097182
G.f. A(x) has the property that the first (n+1) terms of A(x)^(n+1) form the n-th row polynomial R_n(y) of triangle A097181 and satisfy R_n(1/2) = 8^n for all n>=0.
Original entry on oeis.org
1, 7, 21, 21, -63, -231, -15, 1521, 3073, -4319, -29631, -29631, 143361, 489345, -255, -3342591, -6684671, 9454081, 64553985, 64553985, -311689215, -1064175615, -4095, 7266627585, 14533263361, -20553129983, -140345589759, -140345589759, 677648531457, 2313636773889
Offset: 0
A(x) = 1 + 7*x + 21*x^2 + 21*x^3 - 63*x^4 - 231*x^5 - 15*x^6 +-...
For n>=0, the first (n+1) coefficients of A(x)^(n+1) forms the
n-th row polynomial R_n(y) of triangle A097181:
A^1 = {1, _7, 21, 21, -63, -231, -15, 1521, ...}
A^2 = {1, 14, _91, 336, 609, -462, -5469, -9516, ...}
A^3 = {1, 21, 210, _1288, 5103, 11655, 2160, -85590, ...}
A^4 = {1, 28, 378, 3220, _18907, 77280, 199860, 153000, ...}
A^5 = {1, 35, 595, 6475, 49910, _283192, 1175190, 3282870, ...}
A^6 = {1, 42, 861, 11396, 108402, 778596, _4296034, 17959968, ...}
These row polynomials satisfy: R_n(1/2) = 8^n:
8^1 = 1 + 14/2;
8^2 = 1 + 21/2 + 210/2^2;
8^3 = 1 + 28/2 + 378/2^2 + 3220/2^3;
8^4 = 1 + 35/2 + 595/2^2 + 6475/2^3 + 49910/2^4.
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R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 16*x/(1-(1-2*x)^8) )); // G. C. Greubel, Sep 17 2019
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seq(coeff(series(16*x/(1-(1-2*x)^8), x, n+2), x, n), n = 0..30); # G. C. Greubel, Sep 17 2019
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CoefficientList[Series[16*x/(1-(1-2*x)^8), {x,0,30}], x] (* G. C. Greubel, Sep 17 2019 *)
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a(n)=polcoeff(16*x/(1-(1-2*x)^8)+x*O(x^n),n,x)
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def A097194_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P(16*x/(1-(1-2*x)^8)).list()
A097194_list(30) # G. C. Greubel, Sep 17 2019
Showing 1-3 of 3 results.
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