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 1-3 of 3 results.

A165409 Transform of 2^n by the aerated Catalan triangle A165408.

Original entry on oeis.org

1, 2, 4, 10, 24, 56, 136, 328, 784, 1896, 4576, 11008, 26592, 64192, 154752, 373696, 902144, 2176640, 5255424, 12687488, 30621952, 73931392, 178484736, 430845952, 1040176640, 2511199232, 6062209024, 14635617280, 35333443584, 85300015104
Offset: 0

Views

Author

Paul Barry, Sep 17 2009

Keywords

Comments

Hankel transform is A165410.

Links

Crossrefs

Cf. A000108, A000129, A165408, A165410.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 2/(1-4*x+Sqrt(1-8*x^3)) )); // G. C. Greubel, Nov 10 2022
  • Mathematica
    CoefficientList[Series[2/(1-4*x+Sqrt[1-8*x^3]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)
  • SageMath
    def A165408(n,k): return 0 if (n>3*k) else (1+(-1)^(n-k))*(3*k-n+2)*binomial(int((n+k)/2), k)/(4*(k+1))
def A165409(n): return sum(2^k*A165408(n,k) for k in range(n+1))
[A165409(n) for n in range(41)] # G. C. Greubel, Nov 10 2022

Formula

G.f.: 1/(1-2*x-2*x^3*c(2*x^3)) = 2/(1-4*x+sqrt(1-8*x^3)) = (1-4*x-sqrt(1-8*x^3) )/(4*x*(1-2*x-x^2)), c(x) the g.f. of A000108.
G.f.: 1/(1-2*x-2*x^3/(1-2*x^3/(1-2*x^3/(1-2*x^3/(1-... (continued fraction).
a(n) = Sum_{k=ceiling(n/3)..n} 2^k*C((n+k)/2, k)*((3*k-n)/2 + 1)*(1+(-1)^(n-k))/(2*(k+1)).
a(n) = Sum_{k=0..n} 2^k * A165408(n,k).
a(n) = Sum_{k=0..n+1} Pell(n-k+1)*(0^k - 2^((k-2)/2)*A000108((k-2)/3)*(1+2*cos(2*Pi*(k-2)/3))/3).
(n+1)*a(n) = 2(n+1)*a(n-1) + (n+1)*a(n-2) + 4*(2*n-7)*a(n-3) - 8(2*n-7)*a(n-4) - 4*(2*n-7)*a(n-5). - R. J. Mathar, Nov 17 2011
a(n) ~ (4+sqrt(2)) * (1+sqrt(2))^n / 8. - Vaclav Kotesovec, Feb 01 2014

A120730 Another version of Catalan triangle A009766.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 0, 5, 4, 1, 0, 0, 0, 5, 9, 5, 1, 0, 0, 0, 0, 14, 14, 6, 1, 0, 0, 0, 0, 14, 28, 20, 7, 1, 0, 0, 0, 0, 0, 42, 48, 27, 8, 1, 0, 0, 0, 0, 0, 42, 90, 75, 35, 9, 1, 0, 0, 0, 0, 0, 0, 132, 165, 110, 44, 10, 1
Offset: 0

Views

Author

Philippe Deléham, Aug 17 2006, corrected Sep 15 2006

Keywords

Comments

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, ...] DELTA [1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, ...] where DELTA is the operator defined in A084938.
Aerated version gives A165408. - Philippe Deléham, Sep 22 2009
T(n,k) is the number of length n left factors of Dyck paths having k up steps. Example: T(5,4)=4 because we have UDUUU, UUDUU, UUUDU, and UUUUD, where U=(1,1) and D=(1,-1). - Emeric Deutsch, Jun 19 2011
With zeros omitted: 1,1,1,1,2,1,2,3,1,5,4,1,... = A008313. - Philippe Deléham, Nov 02 2011

Examples

			As a triangle, this begins:
  1;
  0,  1;
  0,  1,  1;
  0,  0,  2,  1;
  0,  0,  2,  3,  1;
  0,  0,  0,  5,  4,  1;
  0,  0,  0,  5,  9,  5,  1;
  0,  0,  0,  0, 14, 14,  6,  1;
  ...

Links

Crossrefs

Cf. A000007, A000096, A000108, A001405, A001477, A003161, A005586, A005587.
Cf. A008313, A009766, A039598, A039599, A084938, A099376, A105523, A121725.
Cf. A126087, A128386, A121724, A128387, A129123, A132373, A132374, A132375.
Cf. A132889, A151162, A151254, A151281, A156195, A156361, A156362, A156566.
Cf. A156577, A165408, A357654.

Programs

  • Magma
    A120730:= func< n,k | n gt 2*k select 0 else Binomial(n, k)*(2*k-n+1)/(k+1) >;
[A120730(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Nov 07 2022
  • Maple
    G := 4*z/((2*z-1+sqrt(1-4*z^2*t))*(1+sqrt(1-4*z^2*t))): Gser := simplify(series(G, z = 0, 13)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form  # Emeric Deutsch, Jun 19 2011
# second Maple program:
b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
T:= (n, k)-> b(n, 2*k-n):
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Oct 13 2022
  • Mathematica
    b[x_, y_]:= b[x, y]= If[y<0 || y>x, 0, If[x==0, 1, Sum[b[x-1, y+j], {j, {-1, 1}}] ]];
T[n_, k_] := b[n, 2 k - n];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Oct 21 2022, after Alois P. Heinz *)
T[n_, k_]:= If[n>2*k, 0, Binomial[n, k]*(2*k-n+1)/(k+1)];
Table[T[n, k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 07 2022 *)
  • SageMath
    def A120730(n,k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
flatten([[A120730(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Nov 07 2022

Formula

G.f.: G(t,z) = 4*z/((2*z-1+sqrt(1-4*t*z^2))*(1+sqrt(1-4*t*z^2))). - Emeric Deutsch, Jun 19 2011
Sum_{k=0..n} x^k*T(n,n-k) = A001405(n), A126087(n), A128386(n), A121724(n), A128387(n), A132373(n), A132374(n), A132375(n), A121725(n) for x=1,2,3,4,5,6,7,8,9 respectively. [corrected by Philippe Deléham, Oct 16 2008]
T(2*n,n) = A000108(n); A000108: Catalan numbers.
From Philippe Deléham, Oct 18 2008: (Start)
Sum_{k=0..n} T(n,k)^2 = A000108(n) and Sum_{n>=k} T(n,k) = A000108(k+1).
Sum_{k=0..n} T(n,k)^3 = A003161(n).
Sum_{k=0..n} T(n,k)^4 = A129123(n). (End)
Sum_{k=0..n}, T(n,k)*x^k = A000007(n), A001405(n), A151281(n), A151162(n), A151254(n), A156195(n), A156361(n), A156362(n), A156566(n), A156577(n) for x=0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Feb 10 2009
From G. C. Greubel, Nov 07 2022: (Start)
T(n, k) = 0 if n > 2*k, otherwise binomial(n, k)*(2*k-n+1)/(k+1).
Sum_{k=0..n} (-1)^k*T(n,k) = A105523(n).
Sum_{k=0..n} (-1)^k*T(n,k)^2 = -A132889(n), n >= 1.
Sum_{k=0..floor(n/2)} T(n-k, k) = A357654(n).
T(n, n-1) = A001477(n).
T(n, n-2) = [n=2] + A000096(n-3), n >= 2.
T(n, n-3) = 2*[n<5] + A005586(n-5), n >= 3.
T(n, n-4) = 5*[n<7] - 2*[n=4] + A005587(n-7), n >= 4.
T(2*n+1, n+1) = A000108(n+1), n >= 0.
T(2*n-1, n+1) = A099376(n-1), n >= 1. (End)

A165407 Expansion of 1/(1-x-x^3*c(x^3)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 11, 16, 27, 43, 65, 108, 173, 267, 440, 707, 1105, 1812, 2917, 4597, 7514, 12111, 19196, 31307, 50503, 80380, 130883, 211263, 337284, 548547, 885831, 1417582, 2303413, 3720995, 5965622, 9686617, 15652239, 25130844, 40783083, 65913927
Offset: 0

Views

Author

Paul Barry, Sep 17 2009

Keywords

Comments

Hankel transform is A010892(n+1).
Row sums of A165408.
Number of UF-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are UF-equivalent iff the positions of pattern UF are identical in these paths. This also works for the pattern FU. - Sergey Kirgizov, Apr 08 2018
a(n) is the total number of lattice paths from (0,0) to (n-2*i,i) that do not go above the diagonal x=y using steps in {(1,0), (0,1)}. - Alois P. Heinz, Sep 20 2022

Links

Crossrefs

Cf. A000045, A000108, A001622, A010892, A165408.
Trisections give: A026726, A026671, A026674.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 50); Coefficients(R!( (Sqrt(1-4*x^3) -1+2*x)/(2*x*(1-x-x^2)) )); // G. C. Greubel, Nov 09 2022
  • Maple
    b:= proc(x, y) option remember; `if`(y<=x, `if`(x=0, 1,
      b(x-1, y)+`if`(y>0, b(x, y-1), 0)), 0)
    end:
a:= n-> add(b(n-2*i, i), i=0..n/3):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 20 2022
  • Mathematica
    b[x_, y_]:= b[x, y]= If[y<=x, If[x==0, 1, b[x-1, y] +If[y>0, b[x, y-1], 0]], 0];
a[n_] := Sum[b[n-2*i, i], {i, 0, n/3}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 08 2022, after Alois P. Heinz *)
CoefficientList[Series[(Sqrt[1-4*x^3] -1+2*x)/(2*x*(1-x-x^2)), {x,0,50}], x] (* G. C. Greubel, Nov 09 2022 *)
  • SageMath
    def A165407_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 2/(1-2*x+sqrt(1-4*x^3)) ).list()
A165407_list(50) # G. C. Greubel, Nov 09 2022

Formula

G.f.: 2/(1 - 2*x + sqrt(1-4*x^3)) = 1/(1-x-x^3/(1-x^3/(1-x^3/(1-x^3/(1-.... (continued fraction).
a(n) = Sum_{k=ceiling(n/3)..n} C((n+k)/2,k)*((3*k-n)/2 + 1)*(1+(-1)^(n-k))/(2*(k+1)).
a(n) = Sum_{k=0..n+1} Fibonacci(n-k+1)*(0^k - A000108((k-2)/3)*(1+2*cos(2*Pi*(k-2)/3))/3).
(n+1)*a(n) = (n+1)*a(n-1) + (n+1)*a(n-2) +2*(2*n-7)*a(n-3) -2*(2*n-7)*a(n-4) -2*(2*n-7)*a(n-5). - R. J. Mathar, Nov 15 2011
a(3*n) = A026726(n); a(3*n+1) = A026671(n); a(3*n+2) = A026674(n+1). - Philippe Deléham, Feb 01 2014
Limit_{n->oo} a(n+1)/a(n) = A001622. - Alois P. Heinz, Sep 15 2022
Showing 1-3 of 3 results.

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