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 1-4 of 4 results.

A193729 Mirror of the triangle A193728.

Original entry on oeis.org

1, 1, 2, 3, 10, 8, 9, 42, 64, 32, 27, 162, 360, 352, 128, 81, 594, 1728, 2496, 1792, 512, 243, 2106, 7560, 14400, 15360, 8704, 2048, 729, 7290, 31104, 73440, 103680, 87552, 40960, 8192, 2187, 24786, 122472, 344736, 604800, 677376, 473088, 188416, 32768
Offset: 0

Views

Author

Clark Kimberling, Aug 04 2011

Keywords

Comments

T(n, k) is obtained by reversing the rows of the triangle A193728.
Triangle T(n,k), read by rows, given by [1,2,0,0,0,0,...] DELTA [2,2,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

Examples

			First six rows:
   1;
   1,   2;
   3,  10,    8;
   9,  42,   64,   32;
  27, 162,  360,  352,  128;
  81, 594, 1728, 2496, 1792, 512;

Links

Crossrefs

Cf. A000420, A033999, A081038, A081294, A084938.
Cf. A108981, A133494, A193722, A193728, A247560.

Programs

  • Magma
    function T(n, k) // T = A193729
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return k+1;
  else return 3*T(n-1, k) + 4*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 28 2023
  • Mathematica
    (* First program *)
z = 8; a = 1; b = 2; c = 2; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193728 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A193729 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, k+1, 3*T[n-1,k] + 4*T[n -1, k-1]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 28 2023 *)
  • SageMath
    def T(n, k): # T = A193729
    if (k<0 or k>n): return 0
    elif (n<2): return k+1
    else: return 3*T(n-1, k) + 4*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 28 2023

Formula

Let w(n,k) be the triangle of A193728, then the triangle in this sequence is given by w(n,n-k).
T(n,k) = 4*T(n-1,k-1) + 3*T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-2*x-2*x*y)/(1-3*x-4*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 28 2023: (Start)
T(n, 0) = A133494(n).
T(n, 1) = 2*A081038(n-1).
T(n, n) = A081294(n).
Sum_{k=0..n} T(n, k) = (1/7)*(4*[n=0] + 3*A000420(n)).
Sum_{k=0..n} (-1)^k * T(n, k) = A033999(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*[n=0] + A108981(n-1).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = (1/2)*[n=0] + A247560(n-1).
(End)

A247564 a(n) = 3*a(n-2) - 4*a(n-4) with a(0) = 2, a(1) = 1, a(2) = 3, a(3) = 1.

Original entry on oeis.org

2, 1, 3, 1, 1, -1, -9, -7, -31, -17, -57, -23, -47, -1, 87, 89, 449, 271, 999, 457, 1201, 287, -393, -967, -5983, -4049, -16377, -8279, -25199, -8641, -10089, 7193, 70529, 56143, 251943, 139657, 473713, 194399, 413367, 24569, -654751, -703889, -3617721
Offset: 0

Views

Author

Michael Somos, Sep 20 2014

Keywords

Examples

			G.f. = 2 + x + 3*x^2 + x^3 + x^4 - x^5 - 9*x^6 - 7*x^7 - 31*x^8 - 17*x^9 + ...

Links

Crossrefs

Cf. A247487, A247560, A247563.

Programs

  • Haskell
    a247564 n = a247564_list !! n
a247564_list = [2,1,3,1] ++ zipWith (-) (map (* 3) $ drop 2 a247564_list)
                                        (map (* 4) $ a247564_list)
-- Reinhard Zumkeller, Sep 20 2014
  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((2+x-3*x^2-2*x^3)/(1-3*x^2+4*x^4)));  // G. C. Greubel, Aug 04 2018
  • Maple
    H := (n, a, b) -> hypergeom([a - n/2, b - n/2], [1 - n], 8):
a := n -> `if`(n < 3, [2, 1, 3][n+1], (-1)^iquo(n, 2)*H(n, irem(n, 2), 1/2)):
seq(simplify(a(n)), n=0..42); # Peter Luschny, Sep 03 2019
# second Maple program:
a:= n-> (<<0|1>, <-4|3>>^iquo(n, 2, 'r').<[<2, 3>, <1, 1>][1+r]>)[1,1]:
seq(a(n), n=0..42);  # Alois P. Heinz, Sep 03 2019
  • Mathematica
    CoefficientList[Series[(2+x-3*x^2-2*x^3)/(1-3*x^2+4*x^4), {x,0,60}], x] (* G. C. Greubel, Aug 04 2018 *)
  • PARI
    {a(n) = if( n<0, n=-n; 2^-n, 1) * polcoeff( (2 + x - 3*x^2 - 2*x^3) / (1 - 3*x^2 + 4*x^4) + x * O(x^n), n)};

Formula

G.f.: (2 + x - 3*x^2 - 2*x^3) / (1 - 3*x^2 + 4*x^4).
a(n) = A247487(n) * 3^( n == 1 (mod 4) ) for all n in Z.
a(2*n) = A247563(n). a(2*n + 1) = A247560(n).
0 = a(n)*(+2*a(n+2)) + a(n+1)*(+2*a(n+1) - 8*a(n+2) + a(n+3)) + a(n+2)*(+a(n+2)) for all n in Z.
a(n) = (-1)^floor(n/2)*H(n, n mod 2, 1/2) for n >= 3 where H(n, a, b) = hypergeom([a - n/2, b - n/2], [1 - n], 8). - Peter Luschny, Sep 03 2019

A167435 Hankel transform of A084076.

Original entry on oeis.org

1, 2, -16, -3584, -1114112, -771751936, -68719476736, 50102545854496768, 4999067643975288487936, 1104958199127771065681444864, 363815722265501838229553819942912
Offset: 0

Views

Author

Paul Barry, Nov 03 2009

Keywords

Links

Programs

  • Mathematica
    a[n_] := Sum[Binomial[n + k, 2*k]*(-2)^(n - k), {k, 0, n}]; Table[2^(n^2)*(-1)^n*a[n], {n, 0, 10}] (* G. C. Greubel, Jun 13 2016 *)

Formula

a(n) = 2^(n^2)*(-1)^n*Sum_{k=0..n} C(n+k,2k)*(-2)^(n-k).
a(n) = 2^(n^2)*[x^n](1-2*x)/(1+3*x+4*x^2) = 2^(n^2) *A247560(n).

A247565 a(n) = 5*a(n-1) - 10*a(n-2) + 8*a(n-3) with a(0) = 2, a(1) = a(2) = 3.

Original entry on oeis.org

2, 3, 3, 1, -1, 9, 63, 217, 527, 969, 1311, 1081, 47, -87, 7743, 39961, 121679, 270729, 456543, 548857, 344687, -112791, 380031, 5785561, 24225167, 66310473, 135585183, 208622521, 217744559, 87179049, -72570177, 507315097, 3959709647, 14144835849, 35185603551
Offset: 0

Views

Author

Michael Somos, Sep 20 2014

Keywords

Examples

			G.f. = 2 + 3*x + 3*x^2 + x^3 - x^4 + 9*x^5 + 63*x^6 + 217*x^7 + 527*x^8 + ...

Links

Crossrefs

Cf. A247560, A247564.

Programs

  • Magma
    m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((2-7*x+8*x^2)/(1-5*x+10*x^2-8*x^3))); // G. C. Greubel, Aug 04 2018
  • Mathematica
    CoefficientList[Series[(2-7*x+8*x^2)/(1-5*x+10*x^2-8*x^3), {x, 0, 60}], x] (* or *) LinearRecurrence[{5,-10,8}, {2,3,3}, 60] (* G. C. Greubel, Aug 04 2018 *)
  • PARI
    {a(n) = 2^n + real( (1 + quadgen(-7))^n )};
  • PARI
    Vec((2 - 7*x + 8*x^2) / (1 - 5*x + 10*x^2 - 8*x^3) + O(x^50)) \\ Michel Marcus, Sep 22 2014

Formula

G.f.: (2 - 7*x + 8*x^2) / (1 - 5*x + 10*x^2 - 8*x^3).
(n) = a(-1-n) * 2^(2*n+1) for all n in Z.
a(n) = 2^n + A247560(n) for all n in Z.
a(n) = A247564(n+1) * A247564(n) for all n in Z.
0 = a(n)*(+4*a(n+1) + 2*a(n+2)) + a(n+1)*(-5*a(n+1) + a(n+2)) for all n in Z.
Showing 1-4 of 4 results.

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