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%I #56 Sep 08 2022 08:45:58
%S 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,
%T 945,1053,486,1,20,171,810,2295,3888,3645,1458,1,23,231,1323,4725,
%U 10773,15309,12393,4374,1,26,300,2016,8694,24948,47628,58320,41553,13122
%N Triangular array: the fusion of (x+1)^n and (x+2)^n; see Comments for the definition of fusion.
%C 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
%C ...
%C q(k,x)=t(k,0)*x^k+t(k,1)*x^(k-1)+...+t(k,k-1)*x+t(k,k),
%C ...
%C for k=0,1,2,... The Q-upstep of p is the polynomial given by
%C ...
%C 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.
%C ...
%C 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)).
%C ...
%C 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
%C ...
%C q(n+1,0) .. q(n+1,1)........... q(n+1,n) .... q(n+1,n+1)
%C 0 ......... q(n,0)............. q(n,n-1) .... q(n,n)
%C 0 ......... 0.................. q(n-1,n-2) .. q(n-1,n-1)
%C ...
%C 0 ......... 0.................. q(2,1) ...... q(2,2)
%C 0 ......... 0 ................. q(1,0) ...... q(1,1);
%C here, the polynomial q(k,x) is taken to be
%C 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".
%C ...
%C If s=(s(1),s(2),s(3),...) is a sequence, then the infinite square matrix indicated by
%C s(1)...s(2)...s(3)...s(4)...s(5)...
%C ..0....s(1)...s(2)...s(3)...s(4)...
%C ..0......0....s(1)...s(2)...s(3)...
%C ..0......0.......0...s(1)...s(2)...
%C is the self-fusion matrix of s; e.g., A202453, A202670.
%C ...
%C Example: let p(n,x)=(x+1)^n and q(n,x)=(x+2)^n. Then
%C ...
%C w(0,x) = 1 by definition of W
%C w(1,x) = U(p(0,x)) = U(1) = p(0,0)*q(1,x) = 1*(x+2) = x+2;
%C w(2,x) = U(p(1,x)) = U(x+1) = q(2,x) + q(1,x) = x^2+5x+6;
%C 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;
%C ...
%C 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:
%C 1;
%C 1, 2;
%C 1, 5, 6;
%C 1, 8, 21, 18;
%C ...
%C 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.
%C ...
%C In the following examples, r(P**Q) is the mirror of P**Q, obtained by reversing the rows of P**Q.
%C ...
%C ..P...........Q.........P**Q.......r(P**Q)
%C (x+1)^n.....(x+1)^n.....A081277....A118800 (unsigned)
%C (x+1)^n.....(x+2)^n.....A193722....A193723
%C (x+2)^n.....(x+1)^n.....A193724....A193725
%C (x+2)^n.....(x+2)^n.....A193726....A193727
%C (x+2)^n.....(2x+1)^n....A193728....A193729
%C (2x+1)^n....(x+1)^n.....A038763....A136158
%C (2x+1)^n....(2x+1)^n....A193730....A193731
%C (2x+1)^n,...(x+1)^n.....A193734....A193735
%C ...
%C Continuing, let u denote the polynomial x^n+x^(n-1)+...+x+1, and let Fibo[n,x] denote the n-th Fibonacci polynomial.
%C ...
%C P.............Q.........P**Q.......r(P**Q)
%C Fib[n+1,x]...(x+1)^n....A193736....A193737
%C u.............u.........A193738....A193739
%C u**u..........u**u......A193740....A193741
%C ...
%C Regarding A193722:
%C col 1 ..... A000012
%C col 2 ..... A016789
%C col 3 ..... A081266
%C w(n,n) .... A025192
%C w(n,n-1) .. A081038
%C ...
%C Associated with "upstep" as defined above is "downstep" defined at A193842 in connection with fission.
%H Jinyuan Wang, <a href="/A193722/b193722.txt">Rows n=0..101 of triangle, flattened</a>
%H Clark Kimberling, <a href="https://www.fq.math.ca/Papers1/52-3/Kimberling11132013.pdf">Fusion, Fission, and Factors</a>, Fib. Q., 52 (2014), 195-202.
%F 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
%F 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
%F T(n, k) = 3^(k-1)*( binomial(n-1,k) + 2*binomial(n,k) ). - _G. C. Greubel_, Feb 18 2020
%e First six rows:
%e 1;
%e 1, 2;
%e 1, 5, 6;
%e 1, 8, 21, 18;
%e 1, 11, 45, 81, 54;
%e 1, 14, 78, 216, 297, 162;
%p fusion := proc(p, q, n) local d, k;
%p p(n-1,0)*q(n,x)+add(coeff(p(n-1,x),x^k)*q(n-k,x), k=1..n-1);
%p [1,seq(coeff(%,x,n-1-k), k=0..n-1)] end:
%p p := (n, x) -> (x + 1)^n; q := (n, x) -> (x + 2)^n;
%p A193722_row := n -> fusion(p, q, n);
%p for n from 0 to 5 do A193722_row(n) od; # _Peter Luschny_, Jul 24 2014
%t (* First program *)
%t z = 9; a = 1; b = 1; c = 1; d = 2;
%t p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
%t t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t g[n_] := CoefficientList[w[n, x], {x}]
%t TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193722 *)
%t TableForm[Table[g[n], {n, -1, z}]]
%t Flatten[Table[g[n], {n, -1, z}]] (* A193723 *)
%t (* Second program *)
%t 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 *)
%o (Sage)
%o def fusion(p, q, n):
%o 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))
%o return [1]+[expand(F).coefficient(x,n-1-k) for k in (0..n-1)]
%o A193842_row = lambda k: fusion(lambda n,x: (x+1)^n, lambda n,x: (x+2)^n, k)
%o for n in range(7): A193842_row(n) # _Peter Luschny_, Jul 24 2014
%o (PARI) T(n,k) = 3^(k-1)*(binomial(n-1,k) +2*binomial(n,k)); \\ _G. C. Greubel_, Feb 18 2020
%o (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
%o (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
%Y Cf. A081277, A084938, A118800, A193649, A193723-A193741, A202453.
%K nonn,tabl
%O 0,3
%A _Clark Kimberling_, Aug 04 2011
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