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:
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q(k,x) = t(k,0)*x^k + t(k,1)*x^(k-1) + ... + t(k,k-1)*x + t(k,k),
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for k = 0, 1, 2, ... The Q-downstep of p is the polynomial given by
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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.)
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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)).
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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
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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".
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Example: Let p(n,x) = (x+1)^n and q(n,x) = (x+2)^n. Then
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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.
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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
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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.
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..P........Q...........P^^Q........r(P^^Q)
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Regarding A193842,
col 1 ...... A000012
col 2 ...... A016777
col 3 ...... A081271
w(n,n) ..... A003462
w(n,n-1) ... A014915
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.
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
EXAMPLE
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
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 *)
PROG
(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
(PARI) T(n, k) = sum(j=0, k, 3^(k-j)*binomial(n-j, k-j)); \\ G. C. Greubel, Feb 18 2020
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 07 2011
EXTENSIONS
Name and Comments edited by Petros Hadjicostas, Jun 05 2020
STATUS
approved