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.)

1, 1, 4, 1, 7, 13, 1, 10, 34, 40, 1, 13, 64, 142, 121, 1, 16, 103, 334, 547, 364, 1, 19, 151, 643, 1549, 2005, 1093, 1, 22, 208, 1096, 3478, 6652, 7108, 3280, 1, 25, 274, 1720, 6766, 17086, 27064, 24604, 9841, 1, 28, 349, 2542, 11926, 37384, 78322, 105796

Offset: 0

Author

Clark Kimberling, Aug 07 2011

Keywords

• nonn
• tabl

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-downstep of p is the polynomial given by

...

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.)

...

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)).

...

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

...

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)

...

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".

...

Example: Let p(n,x) = (x+1)^n and q(n,x) = (x+2)^n. Then

...

w(0,x) = 1 by the 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+4*x+4) + 3*(x+2) + 3*1 = x^2 + 7*x + 13,

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.

...

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

...

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.

...

..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

...

Regarding A193842,

col 1 ...... A000012

col 2 ...... A016777

col 3 ...... A081271

w(n,n) ..... A003462

w(n,n-1) ... A014915

Examples

			First six rows, for 0 <= k <= n and 0 <= n <= 5:
  1
  1...4
  1...7....13
  1...10...34....40
  1...13...64....142...121
  1...16...103...334...547...364
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened
• Digital Library of Mathematical Functions, Hypergeometric function, analytic properties.
• Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52(3) (2014), 195-202.

Crossrefs

Cf. A193722 (fusion of P by Q), A193649 (Q-residue), A193843 (mirror of A193842).

Programs

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

Maple

fission := 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);
seq(coeff(%,x,n-k), k=0..n) end:
A193842_row := n -> fission((n,x) -> (x+1)^n, (n,x) -> (x+2)^n, n);
for n from 0 to 5 do A193842_row(n) od; # Peter Luschny, Jul 23 2014
# Alternatively:
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

Mathematica

(* First program *)
z = 10;
p[n_, x_] := (x + 1)^n;
q[n_, x_] := (x + 2)^n
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193842 *)
TableForm[Table[h[n], {n, 0, z}]]  (* A193843 *)
Flatten[Table[h[n], {n, -1, z}]]
(* Second program *)
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 *)

PARI

T(n,k) = sum(j=0,k, 3^(k-j)*binomial(n-j,k-j)); \\ G. C. Greubel, Feb 18 2020

Sage

from mpmath import mp, hyp2f1
mp.dps = 100; mp.pretty = True
def T(n,k):
    return 3^k*binomial(n,k)*hyp2f1(1,-k,-n,1/3)-0^(n-k)//2
for n in range(7):
    print([int(T(n,k)) for k in (0..n)]) # Peter Luschny, Jul 23 2014

Sage

# Second program using the 'fission' operation.
def fission(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))
    return [expand(F).coefficient(x,n-k) for k in (0..n)]
A193842_row = lambda k: fission(lambda n,x: (x+1)^n, lambda n,x: (x+2)^n, k)
for n in range(7): A193842_row(n) # Peter Luschny, Jul 23 2014

Formula

From Peter Bala, Jul 16 2013: (Start)

T(n,k) = Sum_{i = 0..k} 3^(k-i)*binomial(n-i,k-i).

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 + ....

The n-th row polynomial is R(n,x) = (1/(2*x + 1))*((3*x + 1)^(n+1) - x^(n+1)). (End)

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

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

Extensions

Name and Comments edited by Petros Hadjicostas, Jun 05 2020