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A193842 Triangular array: the fission of the polynomial sequence ((x+1)^n: n >= 0) by the polynomial sequence ((x+2)^n: n >= 0). (Fission is defined at Comments.)

%I A193842 #73 Jan 05 2025 19:51:39
%S A193842 1,1,4,1,7,13,1,10,34,40,1,13,64,142,121,1,16,103,334,547,364,1,19,
%T A193842 151,643,1549,2005,1093,1,22,208,1096,3478,6652,7108,3280,1,25,274,
%U A193842 1720,6766,17086,27064,24604,9841,1,28,349,2542,11926,37384,78322,105796
%N A193842 Triangular array: the fission of the polynomial sequence ((x+1)^n: n >= 0) by the polynomial sequence ((x+2)^n: n >= 0). (Fission is defined at Comments.)
%C A193842 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 A193842 ...
%C A193842 q(k,x) = t(k,0)*x^k + t(k,1)*x^(k-1) + ... + t(k,k-1)*x + t(k,k),
%C A193842 ...
%C A193842 for k = 0, 1, 2, ...  The Q-downstep of p is the polynomial given by
%C A193842 ...
%C A193842 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.)
%C A193842 ...
%C A193842 Now suppose that P = (p(n,x): n >= 0) and Q = (q(n,x): n >= 0) 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): n >= 0) of polynomials defined by w(0,x) = 1 and w(n,x) = D(p(n+1,x)).
%C A193842 ...
%C A193842 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
%C A193842 ...
%C A193842 q(n,0) .. q(n,1)............. q(n,n-1) .... q(n,n)
%C A193842 0 ....... q(n-1,0)........... q(n-1,n-2)... q(n-1,n-1)
%C A193842 0 ....... 0.................. q(n-2,n-3) .. q(n-2,n-2)
%C A193842 ...
%C A193842 0 ....... 0.................. q(1,0) ...... q(1,1)
%C A193842 0 ....... 0 ................. 0 ........... q(0,0).
%C A193842 Here, the polynomial q(k,x) is taken to be
%C A193842 q(k,0)*x^k + q(k,1)x^(k-1) + ... + q(k,k)*x + q(k,k);
%C A193842 i.e., "q" is used instead of "t".
%C A193842 ...
%C A193842 Example:  Let p(n,x) = (x+1)^n and q(n,x) = (x+2)^n.  Then
%C A193842 ...
%C A193842 w(0,x) = 1 by the definition of W,
%C A193842 w(1,x) = D(p(2,x)) = 1*(x+2) + 2*1 = x + 4,
%C A193842 w(2,x) = D(p(3,x)) = 1*(x^2+4*x+4) + 3*(x+2) + 3*1 = x^2 + 7*x + 13,
%C A193842 w(3,x) = D(p(4,x)) = 1*(x^3+6*x^2+12*x+8) + 4*(x^2+4x+4) + 6*(x+2) + 4*1 = x^3 + 10*x^2 + 34*x + 40.
%C A193842 ...
%C A193842 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 A193842 1
%C A193842 1...4
%C A193842 1...7....13
%C A193842 1...10...34...40
%C A193842 ...
%C A193842 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.
%C A193842 ...
%C A193842 ..P........Q...........P^^Q........r(P^^Q)
%C A193842 (x+1)^n....(x+2)^n.....A193842.....A193843
%C A193842 (x+1)^n....(x+1)^n.....A193844.....A193845
%C A193842 (x+2)^n....(x+1)^n.....A193846.....A193847
%C A193842 (2x+1)^n...(x+1)^n.....A193856.....A193857
%C A193842 (x+1)^n....(2x+1)^n....A193858.....A193859
%C A193842 (x+1)^n.......u........A054143.....A104709
%C A193842 ..u........(x+1)^n.....A074909.....A074909
%C A193842 ..u...........u........A002260.....A004736
%C A193842 (x+2)^n.......u........A193850.....A193851
%C A193842 ..u.........(x+2)^n....A193844.....A193845
%C A193842 (2x+1)^n......u........A193860.....A193861
%C A193842 ..u.........(2x+1)^n...A115068.....A193862
%C A193842 ...
%C A193842 Regarding A193842,
%C A193842 col 1 ...... A000012
%C A193842 col 2 ...... A016777
%C A193842 col 3 ...... A081271
%C A193842 w(n,n) ..... A003462
%C A193842 w(n,n-1) ... A014915
%H A193842 G. C. Greubel, <a href="/A193842/b193842.txt">Rows n = 0..100 of triangle, flattened</a>
%H A193842 Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/15.2#ii">Hypergeometric function, analytic properties</a>.
%H A193842 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/52-3/Kimberling11132013.pdf">Fusion, Fission, and Factors</a>, Fib. Q., 52(3) (2014), 195-202.
%F A193842 From _Peter Bala_, Jul 16 2013: (Start)
%F A193842 T(n,k) = Sum_{i = 0..k} 3^(k-i)*binomial(n-i,k-i).
%F A193842 O.g.f.: 1/((1 - x*t)*(1 - (1 + 3*x)*t)) = 1 + (1 + 4*x)*t + (1 + 7*x + 13*x^2)*t^2 + ....
%F A193842 The n-th row polynomial is R(n,x) = (1/(2*x + 1))*((3*x + 1)^(n+1) - x^(n+1)). (End)
%F A193842 T(n,k) = T(n-1,k) + 4*T(n-1,k-1) - T(n-2,k-1) - 3*T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 4, T(n,k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Jan 17 2014
%F A193842 T(n,k) = 3^k * C(n,k) * hyp2F1(1, -k, -n, 1/3) with or without the additional term -0^(n-k)/2 depending on the exact definition of the hypergeometric function used. Compare formulas 15.2.5 and 15.2.6 in the DLMF reference. - _Peter Luschny_, Jul 23 2014
%e A193842 First six rows, for 0 <= k <= n and 0 <= n <= 5:
%e A193842   1
%e A193842   1...4
%e A193842   1...7....13
%e A193842   1...10...34....40
%e A193842   1...13...64....142...121
%e A193842   1...16...103...334...547...364
%p A193842 fission := proc(p, q, n) local d, k;
%p A193842 p(n+1,0)*q(n,x)+add(coeff(p(n+1,x),x^k)*q(n-k,x), k=1..n);
%p A193842 seq(coeff(%,x,n-k), k=0..n) end:
%p A193842 A193842_row := n -> fission((n,x) -> (x+1)^n, (n,x) -> (x+2)^n, n);
%p A193842 for n from 0 to 5 do A193842_row(n) od; # _Peter Luschny_, Jul 23 2014
%p A193842 # Alternatively:
%p A193842 p := (n,x) -> add(x^k*(1+3*x)^(n-k),k=0..n): for n from 0 to 7 do [n], PolynomialTools:-CoefficientList(p(n,x), x) od; # _Peter Luschny_, Jun 18 2017
%t A193842 (* First program *)
%t A193842 z = 10;
%t A193842 p[n_, x_] := (x + 1)^n;
%t A193842 q[n_, x_] := (x + 2)^n
%t A193842 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A193842 p1[n_, 0] := p[n, x] /. x -> 0;
%t A193842 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A193842 h[n_] := CoefficientList[d[n, x], {x}]
%t A193842 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A193842 Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193842 *)
%t A193842 TableForm[Table[h[n], {n, 0, z}]]  (* A193843 *)
%t A193842 Flatten[Table[h[n], {n, -1, z}]]
%t A193842 (* Second program *)
%t A193842 Table[SeriesCoefficient[((x+3)^(n+1) -1)/(x+2), {x,0,n-k}], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 18 2020 *)
%o A193842 (Sage)
%o A193842 from mpmath import mp, hyp2f1
%o A193842 mp.dps = 100; mp.pretty = True
%o A193842 def T(n,k):
%o A193842     return 3^k*binomial(n,k)*hyp2f1(1,-k,-n,1/3)-0^(n-k)//2
%o A193842 for n in range(7):
%o A193842     print([int(T(n,k)) for k in (0..n)]) # _Peter Luschny_, Jul 23 2014
%o A193842 (Sage) # Second program using the 'fission' operation.
%o A193842 def fission(p, q, n):
%o A193842     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))
%o A193842     return [expand(F).coefficient(x,n-k) for k in (0..n)]
%o A193842 A193842_row = lambda k: fission(lambda n,x: (x+1)^n, lambda n,x: (x+2)^n, k)
%o A193842 for n in range(7): A193842_row(n) # _Peter Luschny_, Jul 23 2014
%o A193842 (PARI) T(n,k) = sum(j=0,k, 3^(k-j)*binomial(n-j,k-j)); \\ _G. C. Greubel_, Feb 18 2020
%o A193842 (Magma) [ (&+[3^(k-j)*Binomial(n-j,k-j): j in [0..k]]): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Feb 18 2020
%Y A193842 Cf. A193722 (fusion of P by Q), A193649 (Q-residue), A193843 (mirror of A193842).
%K A193842 nonn,tabl
%O A193842 0,3
%A A193842 _Clark Kimberling_, Aug 07 2011
%E A193842 Name and Comments edited by _Petros Hadjicostas_, Jun 05 2020