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A193842 Triangular array: the fission of ((x+1)^n) by ((x+2)^n). (Fission is defined at Comments.) 27
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 (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-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)) 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)).

...

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 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^3+10x^2+34x+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

REFERENCES

C. Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.

LINKS

Table of n, a(n) for n=0..52.

Digital Library of Mathematical Functions, Hypergeometric function, analytic properties

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 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

EXAMPLE

First six rows,for 0<=k<=n, n<=5:

1

1...4

1...7....13

1...10...34....40

1...13...64....142...121

1...16...103...334...547...364

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

MATHEMATICA

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}]]

PROG

(Sage)

from mpmath import *

mp.dps = 100; mp.pretty = True

def T(n, k):

    return 3^k*binomial(n, k)*mpmath.hyp2f1(1, -k, -n, 1/3)-0^(n-k)/2

for n in range(7):

    print [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

CROSSREFS

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

Sequence in context: A050411 A010643 A108906 * A134250 A139045 A262361

Adjacent sequences:  A193839 A193840 A193841 * A193843 A193844 A193845

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 07 2011

STATUS

approved

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Last modified May 23 19:46 EDT 2017. Contains 286926 sequences.