cp's OEIS Frontend

A097189 Row 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.

%I A097189 #30 Aug 29 2024 09:15:35
%S A097189 1,7,55,451,3781,32131,275563,2378971,20640907,179791327,1571002291,
%T A097189 13762897435,120832716655,1062818450155,9363143224315,82600459304203,
%U A097189 729572125425661,6450872644562491,57092964352312951,505729048454449651
%N A097189 Row 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.
%H A097189 G. C. Greubel, <a href="/A097189/b097189.txt">Table of n, a(n) for n = 0..1000</a>
%F A097189 G.f.: A(x) = 3/((1-9*x) + 2*(1-9*x)^(2/3)).
%F A097189 G.f.: A(x) = A004988(x)/(1 - x*A097188(x)).
%F A097189 a(n) = 1 + Sum_{m=0..n-1} Sum_{k=0..n-m} C(k,n-m-k)*3^k*(-1)^(n-m-k)*C(n+k,n). - _Vladimir Kruchinin_, Sep 17 2019
%F A097189 Conjecture: n*(n-1)*a(n) - (19*n-18)*(n-1)*a(n-1) + 9*(11*n^2-31*n+22)*a(n-2) - 9*(3*n-4)*(3*n-5)*a(n-3) = 0. - _R. J. Mathar_, Nov 16 2012
%F A097189 a(n) ~ 3^(2*n+1)/(2*Gamma(2/3) * n^(1/3))*(1 - sqrt(3)*Gamma(2/3)^2 / (4*Pi*n^(1/3))). - _Vaclav Kotesovec_, Feb 04 2014
%p A097189 seq(coeff(series(3/((1-9*x) + 2*(1-9*x)^(2/3)), x, n+1), x, n), n = 0 .. 40); # _G. C. Greubel_, Sep 17 2019
%t A097189 CoefficientList[Series[3/((1-9*x) + 2*(1-9*x)^(2/3)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 04 2014 *)
%o A097189 (PARI) a(n)=polcoeff(3/((1-9*x)+2*(1-9*x+x*O(x^n))^(2/3)),n,x)
%o A097189 (Maxima)
%o A097189 a(n):=sum(sum(binomial(k,n-m-k)*3^k*(-1)^(n-m-k)*binomial(n+k,n),k,0,n-m),m,0,n-1)+1; /* _Vladimir Kruchinin_, Sep 09 2019 */
%o A097189 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 3/((1-9*x) + 2*(1-9*x)^(2/3)) )); // _G. C. Greubel_, Sep 17 2019
%o A097189 (Sage)
%o A097189 def A097189_list(prec):
%o A097189     P.<x> = PowerSeriesRing(QQ, prec)
%o A097189     return P( 3/((1-9*x) + 2*(1-9*x)^(2/3)) ).list()
%o A097189 A097189_list(30) # _G. C. Greubel_, Sep 17 2019
%o A097189 (GAP) List([0..30], n-> 1 + Sum([0..n-1], k-> Sum([0..n-k], j-> (-1)^(n-k-j)*3^j*Binomial(j, n-k-j)*Binomial(n+j, n) )) ); # _G. C. Greubel_, Sep 17 2019
%Y A097189 Cf. A004988, A057083, A097186, A097188.
%K A097189 nonn
%O A097189 0,2
%A A097189 _Paul D. Hanna_, Aug 03 2004