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A192936 Constant term of the reduction by x^2 -> x + 1 of the polynomial p(n,x) = Product_{k=1..n} (x+k).

%I A192936 #24 Sep 08 2022 08:45:58
%S A192936 1,1,3,13,71,463,3497,29975,287265,3042545,35284315,444617525,
%T A192936 6048575335,88347242335,1378930649745,22903345844335,403342641729665,
%U A192936 7506843094993825,147226845692229875,3034786640911840925,65592491119118514375
%N A192936 Constant term of the reduction by x^2 -> x + 1 of the polynomial p(n,x) = Product_{k=1..n} (x+k).
%C A192936 For an introduction to reductions of polynomials by substitutions such as x^2 -> x + 1, see A192232.
%C A192936 Essentially the same as A192239. - _R. J. Mathar_, Aug 10 2011
%H A192936 G. C. Greubel, <a href="/A192936/b192936.txt">Table of n, a(n) for n = 0..100</a>
%F A192936 a(n) = 1/10*(5-sqrt(5))*Gamma(n+3/2+1/2*sqrt(5))/Gamma(3/2+1/2*sqrt(5)) - 1/10*(5+sqrt(5))*Gamma(1/2*sqrt(5)-1/2)*sin(1/2*Pi*(5+sqrt(5))) *Gamma(n+3/2-1/2*sqrt(5))/Pi. - _Vaclav Kotesovec_, Oct 26 2012
%F A192936 a(n) = (-1)^n*Sum_{k=0..n+2} Stirling1(n+2,k)*Fibonacci(k+1). - _G. C. Greubel_, Feb 16 2019
%e A192936 The first four polynomials p(n,x) and their reductions are as follows:
%e A192936 p(0,x) = 1
%e A192936 p(1,x) = (x+1) -> 1 + x
%e A192936 p(2,x) = (x+1)*(x+2) -> 3 + 4*x
%e A192936 p(3,x) = (x+1)*(x+2)*(x+3) -> 13 + 19*x
%e A192936 From these, read
%e A192936 A192936=(1,1,3,13,...) and A192239=(0,1,3,13,...)
%t A192936 (* First program *)
%t A192936 q = x^2; s = x + 1; z = 26;
%t A192936 p[0, x]:= 1; p[n_, x_]:= (x+n)*p[n-1, x];
%t A192936 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192936 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1];
%t A192936 t:= Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192936 Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192936 *)
%t A192936 Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A306183 *)
%t A192936 (* Second program *)
%t A192936 Table[(-1)^n*Sum[StirlingS1[n+2, k]*Fibonacci[k+1], {k, 0, n+2}], {n, 0, 30}] (* _G. C. Greubel_, Feb 16 2019 *)
%o A192936 (PARI) {a(n) = (-1)^n*sum(k=0,n+2, stirling(n+2,k,1)*fibonacci(k+1))};
%o A192936 vector(30, n, n--; a(n)) \\ _G. C. Greubel_, Feb 16 2019
%o A192936 (Magma) [(-1)^n*(&+[StirlingFirst(n+2,k)*Fibonacci(k+1): k in [0..n+2]]): n in [0..30]]; // _G. C. Greubel_, Feb 16 2019
%o A192936 (Sage) [sum((-1)^k*stirling_number1(n+2,k)*fibonacci(k+1) for k in (0..n+2)) for n in (0..30)] # _G. C. Greubel_, Feb 16 2019
%o A192936 (GAP) List([0..30], n-> (-1)^n*Sum([0..n+2], k-> (-1)^(n-k)* Stirling1(n+2, k)*Fibonacci(k+1)) ); # _G. C. Greubel_, Jul 27 2019
%Y A192936 Cf. A192232, A192744, A192239.
%K A192936 nonn
%O A192936 0,3
%A A192936 _Clark Kimberling_, Jul 13 2011