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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.

%I A097182 #13 Sep 08 2022 08:45:14
%S A097182 1,7,21,21,-63,-231,-15,1521,3073,-4319,-29631,-29631,143361,489345,
%T A097182 -255,-3342591,-6684671,9454081,64553985,64553985,-311689215,
%U A097182 -1064175615,-4095,7266627585,14533263361,-20553129983,-140345589759,-140345589759,677648531457,2313636773889
%N 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.
%H A097182 G. C. Greubel, <a href="/A097182/b097182.txt">Table of n, a(n) for n = 0..1000</a>
%F A097182 G.f.: A(x) = 16*x/(1-(1-2*x)^8).
%e A097182 A(x) = 1 + 7*x + 21*x^2 + 21*x^3 - 63*x^4 - 231*x^5 - 15*x^6 +-...
%e A097182 For n>=0, the first (n+1) coefficients of A(x)^(n+1) forms the
%e A097182 n-th row polynomial R_n(y) of triangle A097181:
%e A097182 A^1 = {1, _7,  21,    21,    -63,    -231,      -15,     1521, ...}
%e A097182 A^2 = {1, 14, _91,   336,    609,    -462,    -5469,    -9516, ...}
%e A097182 A^3 = {1, 21, 210, _1288,   5103,   11655,     2160,   -85590, ...}
%e A097182 A^4 = {1, 28, 378,  3220, _18907,   77280,   199860,   153000, ...}
%e A097182 A^5 = {1, 35, 595,  6475,  49910, _283192,  1175190,  3282870, ...}
%e A097182 A^6 = {1, 42, 861, 11396, 108402,  778596, _4296034, 17959968, ...}
%e A097182 These row polynomials satisfy: R_n(1/2) = 8^n:
%e A097182 8^1 = 1 + 14/2;
%e A097182 8^2 = 1 + 21/2 + 210/2^2;
%e A097182 8^3 = 1 + 28/2 + 378/2^2 + 3220/2^3;
%e A097182 8^4 = 1 + 35/2 + 595/2^2 + 6475/2^3 + 49910/2^4.
%p A097182 seq(coeff(series(16*x/(1-(1-2*x)^8), x, n+2), x, n), n = 0..30); # _G. C. Greubel_, Sep 17 2019
%t A097182 CoefficientList[Series[16*x/(1-(1-2*x)^8), {x,0,30}], x] (* _G. C. Greubel_, Sep 17 2019 *)
%o A097182 (PARI) a(n)=polcoeff(16*x/(1-(1-2*x)^8)+x*O(x^n),n,x)
%o A097182 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 16*x/(1-(1-2*x)^8) )); // _G. C. Greubel_, Sep 17 2019
%o A097182 (Sage)
%o A097182 def A097194_list(prec):
%o A097182     P.<x> = PowerSeriesRing(QQ, prec)
%o A097182     return P(16*x/(1-(1-2*x)^8)).list()
%o A097182 A097194_list(30) # _G. C. Greubel_, Sep 17 2019
%Y A097182 Cf. A097181, A097183, A097184, A097185.
%K A097182 sign
%O A097182 0,2
%A A097182 _Paul D. Hanna_, Aug 03 2004