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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, 13122 (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+1)^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)

Fib[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.

LINKS

Jinyuan Wang, Rows n=0..101 of triangle, flattened

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

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. - 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. - Philippe Deléham, Oct 05 2011

T(n, k) = 3^(k-1)*( binomial(n-1,k) + 2*binomial(n,k) ). - G. C. Greubel, Feb 18 2020

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

(* First program *)

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 *)

(* Second program *)

Table[3^(k-1)*(Binomial[n-1, k] +2*Binomial[n, k]), {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 18 2020 *)

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

(PARI) T(n, k) = 3^(k-1)*(binomial(n-1, k) +2*binomial(n, k)); \\ G. C. Greubel, Feb 18 2020

(MAGMA) [3^(k-1)*( Binomial(n-1, k) + 2*Binomial(n, k) ): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020

(GAP) Flat(List([0..10], n-> List([0..n], k-> 3^(k-1)*( Binomial(n-1, k) + 2*Binomial(n, k) ) ))); # G. C. Greubel, Feb 18 2020

CROSSREFS

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

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Last modified May 24 19:14 EDT 2020. Contains 334580 sequences. (Running on oeis4.)