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

%I #24 Sep 08 2022 08:45:58

%S 1,1,3,13,71,463,3497,29975,287265,3042545,35284315,444617525,

%T 6048575335,88347242335,1378930649745,22903345844335,403342641729665,

%U 7506843094993825,147226845692229875,3034786640911840925,65592491119118514375

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

%C For an introduction to reductions of polynomials by substitutions such as x^2 -> x + 1, see A192232.

%C Essentially the same as A192239. - _R. J. Mathar_, Aug 10 2011

%H G. C. Greubel, <a href="/A192936/b192936.txt">Table of n, a(n) for n = 0..100</a>

%F 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 a(n) = (-1)^n*Sum_{k=0..n+2} Stirling1(n+2,k)*Fibonacci(k+1). - _G. C. Greubel_, Feb 16 2019

%e The first four polynomials p(n,x) and their reductions are as follows:

%e p(0,x) = 1

%e p(1,x) = (x+1) -> 1 + x

%e p(2,x) = (x+1)*(x+2) -> 3 + 4*x

%e p(3,x) = (x+1)*(x+2)*(x+3) -> 13 + 19*x

%e From these, read

%e A192936=(1,1,3,13,...) and A192239=(0,1,3,13,...)

%t (* First program *)

%t q = x^2; s = x + 1; z = 26;

%t p[0, x]:= 1; p[n_, x_]:= (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 Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192936 *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A306183 *)

%t (* Second program *)

%t 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 (PARI) {a(n) = (-1)^n*sum(k=0,n+2, stirling(n+2,k,1)*fibonacci(k+1))};

%o vector(30, n, n--; a(n)) \\ _G. C. Greubel_, Feb 16 2019

%o (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 (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 (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 Cf. A192232, A192744, A192239.

%K nonn

%O 0,3

%A _Clark Kimberling_, Jul 13 2011