OFFSET
0,3
COMMENTS
For an introduction to reductions of polynomials by substitutions such as x^2 -> x + 1, see A192232.
Essentially the same as A192239. - R. J. Mathar, Aug 10 2011
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..100
FORMULA
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
a(n) = (-1)^n*Sum_{k=0..n+2} Stirling1(n+2,k)*Fibonacci(k+1). - G. C. Greubel, Feb 16 2019
EXAMPLE
MATHEMATICA
(* First program *)
q = x^2; s = x + 1; z = 26;
p[0, x]:= 1; p[n_, x_]:= (x+n)*p[n-1, x];
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1];
t:= Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192936 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A306183 *)
(* Second program *)
Table[(-1)^n*Sum[StirlingS1[n+2, k]*Fibonacci[k+1], {k, 0, n+2}], {n, 0, 30}] (* G. C. Greubel, Feb 16 2019 *)
PROG
(PARI) {a(n) = (-1)^n*sum(k=0, n+2, stirling(n+2, k, 1)*fibonacci(k+1))};
vector(30, n, n--; a(n)) \\ G. C. Greubel, Feb 16 2019
(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
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 13 2011
STATUS
approved