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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 (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 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

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Last modified November 17 22:58 EST 2018. Contains 317279 sequences. (Running on oeis4.)