login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A192936 Constant term of the reduction by x^2 -> x + 1 of the polynomial p(n,x) = Product_{k=1..n} (x+k). 2
1, 1, 3, 13, 71, 463, 3497, 29975, 287265, 3042545, 35284315, 444617525, 6048575335, 88347242335, 1378930649745, 22903345844335, 403342641729665, 7506843094993825, 147226845692229875, 3034786640911840925, 65592491119118514375 (list; graph; refs; listen; history; text; internal format)
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

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

p(0,x) = 1

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

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

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

From these, read

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

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

Cf. A192232, A192744, A192239.

Sequence in context: A233824 A003319 A192239 * A000261 A111140 A302699

Adjacent sequences:  A192933 A192934 A192935 * A192937 A192938 A192939

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 13 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 26 09:32 EDT 2020. Contains 334620 sequences. (Running on oeis4.)