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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-downstep of p is the polynomial given by
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D(p)=p(n)*q(n-1,x)+p(n-1)*q(n-2,x)+...+p(1)*q(0,x); note that p(0) does not appear. ("Q-downstep" as just defined differs slightly from "Q-downstep" as defined for a different purpose at A193649.)
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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 fission 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,x)=D(p(n+1,x)).
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Strictly speaking, ^^ is an operation on sequences of polynomials. However, if P and Q are regarded as numerical triangles (of coefficients of polynomials), then ^^ can be regarded as an operation on numerical triangles. In this case, row n of P^^Q, for n>0, is given by the matrix product P(n+1)*QQ(n), where P(n+1)=(p(n+1,n+1)...p(n+1,n)......p(n+1,2), p(n+1,1)) and QQ(n) is the (n+1)-by-(n+1) matrix given by
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q(n,0) .. q(n,1)............. q(n,n-1) .... q(n,n)
0 ....... q(n-1,0)........... q(n-1,n-2)... q(n-1,n-1)
0 ....... 0.................. q(n-2,n-3) .. q(n-2,n-2)
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0 ....... 0.................. q(1,0) ...... q(1,1)
0 ....... 0 ................. 0 ........... q(0,0));
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);
i.e., "q" is used instead of "t".
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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)=D(p(2,x))=1*(x+2)+2*1=x+4
w(2,x)=D(p(3,x))=1*(x^2+4x+4)+3*(x+2)+3*1=x^2+7x+13
w(3,x)=D(p(4,x))=1*(x^3+6x^2+12x+8)+4*(x^2+4x+4)+6*(x+2)+4*1=x^4+10x^2+34x+40.
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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...4
1...7....13
1...10...34...40
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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. Let u denote the polynomial x^n+x^n-1+...+x+1.
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..P........Q...........P**Q........r(P**Q)
(x+1)^n....(x+2)^n.....A193842.....A193843
(x+1)^n....(x+1)^n.....A193844.....A193845
(x+2)^n....(x+1)^n.....A193846.....A193847
(2x+1)^n...(x+1)^n.....A193856.....A193857
(x+1)^n....(2x+1)^n....A193858.....A193859
(x+1)^n.......u........A054143.....A104709
..u........(x+1)^n.....A074909.....A074909
..u...........u........A002260.....A004736
(x+2)^n.......u........A193850.....A193851
..u.........(x+2)^n....A193844.....A193845
(2x+1)^n......u........A193860.....A193861
..u.........(2x+1)^n...A115068.....A193862
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Regarding A193842,
col 1 ...... A000012
col 2 ...... A016777
col 3 ...... A081271
w(n,n) ..... A003462
w(n,n-1) ... A014915
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