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%I #13 Sep 08 2022 08:45:58
%S 0,1,5,31,224,1844,17028,174284,1958176,23959760,317128240,4514617360,
%T 68784608640,1116787186240,19248968150720,351024831699520,
%U 6752328440253440,136640443206206720,2901703626188801280,64522443639953657600
%N a(n) = A192942(n)/2.
%C See A192942.
%H G. C. Greubel, <a href="/A192950/b192950.txt">Table of n, a(n) for n = 0..440</a>
%F a(n) = 1/10*sqrt(5)*Gamma(n+2+sqrt(5))/Gamma(sqrt(5)+2) - 1/10*sin(Pi*(sqrt(5)+3))*Gamma(sqrt(5)+1)*Gamma(n+2-sqrt(5))/(Pi*(sqrt(5)-1)). - _Vaclav Kotesovec_, Oct 26 2012
%e (See A192942.)
%t (* First program *)
%t q = x^2; s = x + 1; z = 26;
%t p[0, x]:= 1;
%t p[n_, x_]:= (2*x + n)*p[n-1, x];
%t Table[Expand[p[n, x]], {n, 0, 7}]
%t reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192941 *)
%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192942 *)
%t u2/2 (* A192950 *)
%t (* Additional programs *)
%t With[{s = Sqrt[5]}, Table[FullSimplify[(s*Gamma[n+2+s]/Gamma[s+2] - Sin[Pi*(s+3)]*Gamma[s+1]*Gamma[n+2-s]/(Pi*(s-1)))/10], {n, 0, 20}]] (* _G. C. Greubel_, Jul 25 2019 *)
%o (PARI) default(realprecision, 100); vector(20, n, n--; s=sqrt(5); round(s*gamma(n+2+s)/gamma(s+2) - sin(Pi*(s+3))*gamma(s+1)*gamma(n+2-s)/(Pi*(s-1)))/10 ) \\ _G. C. Greubel_, Jul 25 2019
%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); s:=Sqrt(5); [Round(s*Gamma(n+2+s)/Gamma(s+2) - Sin(Pi(R)*(s+3))*Gamma(s+1) *Gamma(n+2-s)/(Pi(R)*(s-1)))/10: n in [0..20]]; // _G. C. Greubel_, Jul 25 2019
%o (Sage) s=sqrt(5); [round(s*gamma(n+2+s)/gamma(s+2) - sin(pi*(s+3))* gamma(s+1)*gamma(n+2-s)/(pi*(s-1)))/10 for n in (0..20)] # _G. C. Greubel_, Jul 25 2019
%Y Cf. A192942.
%K nonn
%O 0,3
%A _Clark Kimberling_, Jul 13 2011
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