login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A193722 Triangular array: the fusion of (x+1)^n and (x+2)^n; see Comments for the definition of fusion. 93
1, 1, 2, 1, 5, 6, 1, 8, 21, 18, 1, 11, 45, 81, 54, 1, 14, 78, 216, 297, 162, 1, 17, 120, 450, 945, 1053, 486, 1, 20, 171, 810, 2295, 3888, 3645, 1458, 1, 23, 231, 1323, 4725, 10773, 15309, 12393, 4374, 1, 26, 300, 2016, 8694, 24948, 47628, 58320, 41553 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Suppose that p=p(n)*x^n+p(n-1)*x^(n-1)+...+p(1)*x+p(0) is a polynomial 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-upstep of p is the polynomial given by

...

U(p)=p(n)*q(n+1,x)+p(n-1)*q(n,x)+...+p(0)*q(1,x); note that q(0,x) does not appear.

...

Now suppose that P=(p(n,x)) and Q=(q(n,x)) are sequences of polynomials, where n indicates degree.  The fusion of P by Q, denoted by P**Q, is introduced here as the sequence W=(w(n,x)) of polynomials defined by w(0,x)=1 and w(n+1,x)=U(p(n,x)).

...

Strictly speaking, ** is an operation on sequences of polynomials.  However, if P and Q are regarded as numerical triangles (e.g., coefficients of polynomials), then ** can be regarded as an operation on numerical triangles.  In this case, row (n+1) of P**Q, for n>=0, is given by the matrix product P(n)*QQ(n), where P(n)=(p(n,n)...p(n,n-1)......p(n,1), p(n,0)) and QQ(n) is the (n+1)-by-(n+2) matrix given by

...

q(n+1,0) .. q(n+1,1)........... q(n+1,n) .... q(n+1,n+1)

0 ......... q(n,0)............. q(n,n-1) .... q(n,n)

0 ......... 0.................. q(n-1,n-2) .. q(n-1,n-1)

...

0 ......... 0.................. q(2,1) ...... q(2,2)

0 ......... 0 ................. q(1,0) ...... q(1,1);

here, the polynomial q(k,x) is taken to be

q(k,0)*x^k + q(k,1)x^(k-1) + ... + q(k,k)*x+q(k,k-1); i.e., "q" is used instead of "t".

...

If s=(s(1),s(2),s(3),...) is a sequence, then the infinite square matrix indicated by

s(1)...s(2)...s(3)...s(4)...s(5)...

..0....s(1)...s(2)...s(3)...s(4)...

..0......0....s(1)...s(2)...s(3)...

..0......0.......0...s(1)...s(2)...

is the self-fusion matrix of s; e.g., A202453, A202670.

...

Example:  let p(n,x)=(x+1)^n and q(n,x)=(x+2)^n.  Then

  ...

w(0,x)=1 by definition of W

w(1,x)=U(p(0,x))=U(1)=p(0,0)q(1,x)=1*(x+2)=x+2

w(2,x)=U(p(1,x))=U(x+1)=q(2,x)+q(1,x)=x^2+5x+6

w(3,x)=U(p(2,x))=U(x^2+2x+1)=q(3,x)+2q(2,x)+q(1,x)=x^3+8x^2+21x+18.

  ...

From these first 4 polynomials in the sequence P**Q, we can write the first 4 rows of P**Q when P, Q, and P**Q are regarded as triangles:

1

1...2

1...5...6

1...8...21...18

...

Generally, if P and Q are the sequences given by p(n,x)=(ax+b)^n and q(n,x)=(cx+d)^n, then P**Q is given by (cx+d)(bcx+a+bd)^n.

...

In the following examples, r(P**Q) is the mirror of P**Q, obtained by reversing the rows of P**Q.

...

..P...........Q.........P**Q.......r(P**Q)

(x+1)^n.....(x+1)^n.....A081277....A118800 (unsigned)

(x+1)^n.....(x+2)^n.....A193722....A193723

(x+2)^n.....(x+1)^n.....A193724....A193725

(x+2)^n.....(x+2)^n.....A193726....A193727

(x+2)^n.....(2x+1)^n....A193728....A193729

(2x+)^n.....(x+1)^n.....A038763....A136158

(2x+1)^n....(2x+1)^n....A193730....A193731

(2x+1)^n,...(x+1)^n.....A193734....A193735

...

Continuing, let u denote the polynomial x^n+x^(n-1)+...+x+1, and let Fibo[n,x] denote the n-th Fibonacci polynomial.

...

P.............Q.........P**Q.......r(P**Q)

Fibo[n+1,x]..(x+1)^n....A193736....A193737

u.............u.........A193738....A193739

u**u..........u**u......A193740....A193741

...

Regarding A193722:

col 1 ..... A000012

col 2 ..... A016789

col 3 ..... A081266

w(n,n) .... A025192

w(n,n-1) .. A081038

...

Associated with "upstep" as defined above is "downstep" defined at A193842 in connection with fission.

REFERENCES

C. Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.

LINKS

Table of n, a(n) for n=0..53.

FORMULA

Triangle T(n,k), read by rows, given by [1,0,0,0,0,0,0,0,...] DELTA [2,1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. [From Philippe Deléham, Oct 04 2011]

T(n,k)=3*T(n-1,k-1)+T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2 - From Philippe Deléham, Oct 05 2011.

EXAMPLE

First six rows:

1

1....2

1....5....6

1....8....21...18

1....11...45...81...54

1....14...78...216..297..162

MAPLE

fusion := proc(p, q, n) local d, k;

p(n-1, 0)*q(n, x)+add(coeff(p(n-1, x), x^k)*q(n-k, x), k=1..n-1);

[1, seq(coeff(%, x, n-1-k), k=0..n-1)] end:

p := (n, x) -> (x + 1)^n; q := (n, x) -> (x + 2)^n;

A193722_row := n -> fusion(p, q, n);

for n from 0 to 5 do A193722_row(n) od; # Peter Luschny, Jul 24 2014

MATHEMATICA

z = 9; a = 1; b = 1; c = 1; d = 2;

p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n

t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;

w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1

g[n_] := CoefficientList[w[n, x], {x}]

TableForm[Table[Reverse[g[n]], {n, -1, z}]]

Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193722 *)

TableForm[Table[g[n], {n, -1, z}]]

Flatten[Table[g[n], {n, -1, z}]] (* A193723 *)

PROG

(Sage)

def fusion(p, q, n):

    F = p(n-1, 0)*q(n, x)+add(expand(p(n-1, x)).coefficient(x, k)*q(n-k, x) for k in (1..n-1))

    return [1]+[expand(F).coefficient(x, n-1-k) for k in (0..n-1)]

A193842_row = lambda k: fusion(lambda n, x: (x+1)^n, lambda n, x: (x+2)^n, k)

for n in range(7): A193842_row(n) # Peter Luschny, Jul 24 2014

CROSSREFS

Cf. A084938, A193649, A193723-A193741, A081277, A118800, A202453.

Sequence in context: A269019 A184234 A193816 * A193635 A241168 A145324

Adjacent sequences:  A193719 A193720 A193721 * A193723 A193724 A193725

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 04 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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 29 15:10 EDT 2017. Contains 284273 sequences.