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A306183 The coefficients of x in the reduction of x^2 -> x + 1 for the polynomial p(n,x) = Product_{k=1..n} (x+k).

%I A306183 #33 Sep 08 2022 08:46:21
%S A306183 0,1,4,19,108,719,5496,47465,457160,4858865,56490060,713165035,
%T A306183 9715762980,142069257055,2219386098160,36889108220305,650018185589520,
%U A306183 12103669982341025,237476572759473300,4896758300881695875,105866710959427454300,2394660132226522508975,56560492065670933962600
%N A306183 The coefficients of x in the reduction of x^2 -> x + 1 for the polynomial p(n,x) = Product_{k=1..n} (x+k).
%C A306183 See A192936 for the constant term of the reduction x^2 -> x + 1 for the polynomial p(n,x) = Product_{k=1..n} (x+k).
%H A306183 G. C. Greubel, <a href="/A306183/b306183.txt">Table of n, a(n) for n = 0..445</a>
%F A306183 a(n) = (-1)^(n+1)*Sum_{k=0..n+2} Stirling1(n+2,k)*A000045(k).
%F A306183 From _Vaclav Kotesovec_, Feb 09 2019: (Start)
%F A306183 a(n) = 2*n*a(n-1) - (n^2 - n - 1)*a(n-2).
%F A306183 a(n) = cos(Pi*sqrt(5)/2) * (Gamma(sqrt(5)*phi) * Gamma(n + 1/phi^2) / phi^2 - phi^2 * Gamma(sqrt(5)/phi) * Gamma(n + phi^2)) / (Pi*sqrt(5)).
%F A306183 a(n) ~ c * n! * n^phi, where c = -cos(sqrt(5)*Pi/2) * (5 + 3*sqrt(5)) * Gamma((5 - sqrt(5))/2) / (10*Pi) = 0.30858712435869... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. (End)
%t A306183 Table[(-1)^(n+1)*Sum[StirlingS1[n+2,k]*Fibonacci[k],{k,0,n+2}],{n,0,30}]
%o A306183 (PARI) {a(n) = (-1)^(n+1)*sum(k=0,n+2, stirling(n+2,k,1)*fibonacci(k))};
%o A306183 vector(30, n, n--; a(n))
%o A306183 (Magma) [(-1)^(n+1)*(&+[StirlingFirst(n+2,k)*Fibonacci(k): k in [0..n+2]]): n in [0..30]];
%o A306183 (Sage) [sum((-1)^(k+1)*stirling_number1(n+2,k)*fibonacci(k) for k in (0..n+2)) for n in (0..30)]
%Y A306183 Cf. A192936, A323620 (signed)
%K A306183 nonn
%O A306183 0,3
%A A306183 _G. C. Greubel_, Feb 07 2019