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A193649 Q-residue of the (n+1)st Fibonacci polynomial, where Q is the triangular array (t(i,j)) given by t(i,j)=1.  (See Comments.) 19
1, 1, 3, 5, 15, 33, 91, 221, 583, 1465, 3795, 9653, 24831, 63441, 162763, 416525, 1067575, 2733673, 7003971, 17938661, 45954543, 117709185, 301527355, 772364093, 1978473511 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Suppose that p=p(0)*x^n+p(1)*x^(n-1)+...+p(n-1)*x+p(n) is a polynomial of positive degree and that Q is a sequence of polynomials: q(k,x)=t(k,0)*x^k+t(k,1)*x^(k-1)+...+t(k,k-1)*x+t(k,k), for k=0,1,2,...  The Q-downstep of p is the polynomial given by D(p)=p(0)*q(n-1,x)+p(1)*q(n-2,x)+...+p(n-1)*q(0,x)+p(n).

Since degree(D(p))<degree(p), the result of n applications of D is a constant, which we call the Q-residue of p.  If p is a constant to begin with, we define D(p)=p.

Example: let p(x)=2*x^3+3*x^2+4*x+5 and q(k,x)=(x+1)^k.

  D(p)=2(x+1)^2+3(x+1)+4(1)+5=2x^2+7x+14

  D(D(p))=2(x+1)+7(1)+14=2x+23

  D(D(D(p)))=2(1)+23=25;

  the Q-residue of p is 25.

We may regard the sequence Q of polynomials as the triangular array formed by coefficients:

t(0,0)

t(1,0)....t(1,1)

t(2,0)....t(2,1)....t(2,2)

t(3,0)....t(3,1)....t(3,2)....t(3,3)

  and regard p as the vector (p(0),p(1),...,p(n)).  If P is a sequence of polynomials [or triangular array having (row n)=(p(0),p(1),...,p(n))], then the Q-residues of the polynomials form a numerical sequence.

Following are examples in which Q is the triangle given by t(i,j)=1 for 0<=i<=j:

Q.....P...................Q-residue of P

1.....1...................A000079, 2^n

1....(x+1)^n..............A007051, (1+3^n)/2

1....(x+2)^n..............A034478, (1+5^n)/2

1....(x+3)^n..............A034494, (1+7^n)/2

1....(2x+1)^n.............A007582

1....(3x+1)^n.............A081186

1....(2x+3)^n.............A081342

1....(3x+2)^n.............A081336

1.....A040310.............A193649

1....(x+1)^n+(x-1)^n)/2...A122983

1....(x+2)(x+1)^(n-1).....A057198

1....(1,2,3,4,...,n)......A002064

1....(1,1,2,3,4,...,n)....A048495

1....(n,n+1,...,2n).......A087323

1....(n+1,n+2,...,2n+1)...A099035

1....p(n,k)=(2^(n-k))*3^k.A085350

1....p(n,k)=(3^(n-k))*2^k.A090040

1....A008288 (Delannoy)...A193653

1....A054142..............A101265

1....cyclotomic...........A193650

1....(x+1)(x+2)...(x+n)...A193651

1....A114525..............A193662

More examples:

Q...........P.............Q-residue of P

(x+1)^n...(x+1)^n.........A000110, Bell numbers

(x+1)^n...(x+2)^n.........A126390

(x+2)^n...(x+1)^n.........A028361

(x+2)^n...(x+2)^n.........A126443

(x+1)^n.....1.............A005001

(x+2)^n.....1.............A193660

A094727.....1.............A193657

(k+1).....(k+1)...........A001906 (even-ind. Fib. nos.)

(k+1).....(x+1)^n.........A112091

(x+1)^n...(k+1)...........A029761

(k+1)......A049310........A193663

(In these last four, (k+1) represents the triangle t(n,k)=k+1, 0<=k<=n.)

A051162...(x+1)^n.........A193658

A094727...(x+1)^n.........A193659

A049310...(x+1)^n.........A193664

A075362....A075362........A193665

Changing the notation slightly leads to the Mathematica program below and the following formulation for the Q-downstep of p:  first, write t(n,k) as q(n,k).  Define r(k)=Sum{q(k-1,i)*r(k-1-i) : i=0,1,...,k-1} Then row n of D(p) is given by v(n)=Sum{p(n,k)*r(n-k) : k=0,1,...,n}.

LINKS

Table of n, a(n) for n=0..24.

FORMULA

Conjecture: G.f.: -(1+x)*(2*x-1) / ( (x-1)*(4*x^2+x-1) ). - R. J. Mathar, Feb 19 2015

EXAMPLE

First five rows of Q, coefficients of Fibonacci polynomials (A049310):

1

1...0

1...0...1

1...0...2...0

1...0...3...0...1

To obtain a(4)=15, downstep four times:

D(x^4+3*x^2+1)=(x^3+x^2+x+1)+3(x+1)+1: (1,1,4,5) [coefficients]

DD(x^4+3*x^2+1)=D(1,1,4,5)=(1,2,11)

DDD(x^4+3*x^2+1)=D(1,2,11)=(1,14)

DDDD(x^4+3*x^2+1)=D(1,14)=15.

MATHEMATICA

q[n_, k_] := 1;

r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];

f[n_, x_] := Fibonacci[n + 1, x];

p[n_, k_] := Coefficient[f[n, x], x, k]; (* A049310 *)

v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]

Table[v[n], {n, 0, 24}]    (* A193649 *)

TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]

Table[r[k], {k, 0, 8}]  (* 2^k *)

TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]]

CROSSREFS

Cf. A192872 (polynomial reduction), A193091 (polynomial augmentation), A193722 (the upstep operation and fusion of polynomial sequences or triangular arrays).

Sequence in context: A018470 A120748 A182143 * A186341 A262326 A148499

Adjacent sequences:  A193646 A193647 A193648 * A193650 A193651 A193652

KEYWORD

nonn

AUTHOR

Clark Kimberling, Aug 02 2011

STATUS

approved

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Last modified April 25 00:46 EDT 2017. Contains 285346 sequences.