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
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q(k,x)=t(k,0)*x^k+t(k,1)*x^(k-1)+...+t(k,k-1)*x+t(k,k),
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for k=0,1,2,... The Q-upstep of p is the polynomial given by
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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.
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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)).
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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
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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)
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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".
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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)...
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Example: let p(n,x)=(x+1)^n and q(n,x)=(x+2)^n. Then
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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;
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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;
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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.
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In the following examples, r(P**Q) is the mirror of P**Q, obtained by reversing the rows of P**Q.
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..P...........Q.........P**Q.......r(P**Q)
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Continuing, let u denote the polynomial x^n+x^(n-1)+...+x+1, and let Fibo[n,x] denote the n-th Fibonacci polynomial.
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P.............Q.........P**Q.......r(P**Q)
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Regarding A193722:
col 1 ..... A000012
col 2 ..... A016789
col 3 ..... A081266
w(n,n) .... A025192
w(n,n-1) .. A081038
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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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 04 2011
STATUS
approved