cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A193857 Mirror of the triangle A193856.

Original entry on oeis.org

1, 5, 1, 19, 8, 1, 65, 43, 11, 1, 211, 194, 76, 14, 1, 665, 793, 422, 118, 17, 1, 2059, 3044, 2059, 776, 169, 20, 1, 6305, 11191, 9221, 4387, 1283, 229, 23, 1, 19171, 39878, 38854, 22382, 8236, 1970, 298, 26, 1, 58025, 138805, 156440, 106000, 47090
Offset: 0

Views

Author

Clark Kimberling, Aug 07 2011

Keywords

Comments

A193857 is obtained by reversing the rows of the triangle A193856.

Examples

			First six rows:
1
5.....1
19....8.....1
65....43....11....1
211...194...76....14....1
665...793...422...118...17...1

Crossrefs

Cf. A193856.

Programs

  • Mathematica
    z = 10;
p[n_, x_] := (2 x + 1)^n;
q[n_, x_] := (x + 1)^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}]]  (* A193856 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193857 *)

Formula

Write w(n,k) for the triangle at A193856. The triangle at A193857 is then given by w(n,n-k).

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.)

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Aug 07 2011

Keywords

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

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
Showing 1-2 of 2 results.

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