A097180 - OEIS

%I #14 Sep 08 2022 08:45:14

%S 1,7,52,395,3036,23506,182904,1428387,11185900,87789702,690212744,

%T 5434455182,42841215704,338081920260,2670388231152,21109070463267,

%U 166980248599884,1321686452484286,10467203182893800,82936871755938970

%N Row sums of triangle A097179, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A077860(y)^(n+1), where R_n(1/2) = 4^n for all n>=0.

%H G. C. Greubel, <a href="/A097180/b097180.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: A(x) = 2/((1-8*x) + (1-8*x)^(3/4)).

%F Conjecture: n*(n-1)*(n+1)*a(n) -12*n*(2*n-1)*(n-1)*a(n-1) +12*(n-1) * (16*n^2-32*n+17)*a(n-2) -16*(4*n-5)*(4*n-7)*(2*n-3)*a(n-3) = 0. - _R. J. Mathar_, Nov 16 2012

%F a(n) ~ 2^(3*n+1) / (Gamma(3/4)*n^(1/4)) * (1 - Gamma(3/4) / (n^(1/4) * sqrt(Pi))). - _Vaclav Kotesovec_, Feb 04 2014

%p seq(coeff(series(2/((1-8*x) + (1-8*x)^(3/4)), x, n+1), x, n), n = 0 ..20); # _G. C. Greubel_, Sep 17 2019

%t CoefficientList[Series[2/((1-8*x) + (1-8*x)^(3/4)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 04 2014 *)

%o (PARI) a(n)=polcoeff(2/((1-8*x)+(1-8*x+x*O(x^n))^(3/4)),n,x)

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 2/((1-8*x) + (1-8*x)^(3/4)) )); // _G. C. Greubel_, Sep 17 2019

%o (Sage)

%o def A097180_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P(2/((1-8*x) + (1-8*x)^(3/4))).list()

%o A097180_list(20) # _G. C. Greubel_, Sep 17 2019

%Y Cf. A077860, A097179.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 02 2004