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.

A166692 Triangle T(n,k) read by rows: T(n,k) = 2^(k-1), k>0, T(n,0) = (n+1) mod 2.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 0, 1, 2, 4, 1, 1, 2, 4, 8, 0, 1, 2, 4, 8, 16, 1, 1, 2, 4, 8, 16, 32, 0, 1, 2, 4, 8, 16, 32, 64, 1, 1, 2, 4, 8, 16, 32, 64, 128, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
Offset: 0

Views

Author

Paul Curtz, Oct 18 2009

Keywords

Comments

Variant of A166918.

Examples

			Triangle begins as:
  1;
  0, 1;
  1, 1, 2;
  0, 1, 2, 4;
  1, 1, 2, 4, 8;
  0, 1, 2, 4, 8, 16;

Links

Crossrefs

Cf. A005578, A011782, A051049, A131577, A106624, A166494, A166918.

Programs

  • Magma
    A166692:= func< n,k | k eq 0 select ((n+1) mod 2) else 2^(k-1) >;
[A166692(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 24 2023
  • Mathematica
    Join[{1,0},Flatten[Riffle[Table[2^Range[0,n],{n,0,10}],{1,0}]]] (* Harvey P. Dale, Jan 18 2015 *)
  • SageMath
    def A166692(n,k): return ((n+1)%2) if (k==0) else 2^(k-1)
flatten([[A166692(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Apr 24 2023

Formula

T(2n, k) = A011782(k).
T(2n+1, k) = A131577(k).
Sum_{k=0..n} T(n,k) = A051049(n).
From G. C. Greubel, Apr 24 2023: (Start)
T(2*n, n) = A011782(n).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n * A005578(n).
Sum_{k=0..n} T(n-k, k) = A106624(n). (End)

Extensions

More terms from Harvey P. Dale, Jan 18 2015

A166866 Interleave the fractals A158405, A002260.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 3, 5, 1, 2, 3, 1, 3, 5, 7, 1, 2, 3, 4, 1, 3, 5, 7, 9, 1, 2, 3, 4, 5, 1, 3, 5, 7, 9, 11, 1, 2, 3, 4, 5, 6, 1, 3, 5, 7, 9, 11, 13, 1, 2, 3, 4, 5, 6, 7, 1, 3, 5, 7, 9, 11, 13, 15, 1, 2, 3, 4, 5, 6, 7, 8, 1, 3, 5, 7, 9, 11, 13, 15, 17, 1, 2, 3, 4, 5, 6, 7, 8, 9
Offset: 0

Views

Author

Paul Curtz, Oct 22 2009

Keywords

Comments

See A166494. Consider a(n+1) sum by 1,1,2,2,3,3=A008619 terms: 1; 1; 3+1=4; 2+1=3; 3+5+1=9; 2+3+1=6; 3+5+7+1=16; 2+3+4+1=10; this is A123596(n+2).Via A001105(n+1),A123596 is linked to (extended) Mendeleyev-Moseley-Seaborg table of the elements.

Crossrefs

Cf. A001105, A002260, A008619, A123596, A158405, A166494.

A383874 a(n) = (3*n+1)!*(3*n)!/((2*n)!*((n+1)!)^2).

Original entry on oeis.org

1, 18, 4200, 3175200, 5137292160, 14544244915200, 64008493310361600, 405192226643043840000, 3493057136053143859200000, 39378260464472988708249600000, 562659674639968187756457984000000, 9940535265182157971578474463232000000, 212816707229761791940688046273331200000000
Offset: 0

Views

Author

Karol A. Penson, May 22 2025

Keywords

Links

Crossrefs

Cf. A166149, A166771, A166494, A166384, A166750, A271049, A064352.

Programs

  • Mathematica
    A383874[n_] := (3*n+1)!*(3*n)!/((2*n)!*((n+1)!)^2);
Array[A383874, 15, 0] (* Paolo Xausa, May 26 2025 *)
  • PARI
    a(n) = (3*n+1)!*(3*n)!/((2*n)!*((n+1)!)^2); \\ Michel Marcus, May 22 2025

Formula

O.g.f.: hypergeom([1/3, 2/3, 2/3, 1, 1, 4/3], [1/2, 2, 2], (729*x)/4).
E.g.f.: hypergeom([1/3, 2/3, 2/3, 1, 1, 4/3], [1/2, 2, 2, 1], (729*x)/4).
a(n) = Integral_{x>=0} x^n*W(x)*dx, n>=0, with W(x) = MeijerG([[],[-1/2,1,1]],[[0,-1/3,-1/3,1/3,-2/3],[]],4*x/729)/(81*Pi^(3/2)), where MeijerG is the Meijer G - function. Apparently W(x) cannot be represented by any other simpler functions. W(x) is a positive function on (0,oo), is singular at x = 0 and goes monotonically to zero as x -> oo. Thus a(n) is a positive definite sequence.
W(x) is the solution of the Stieltjes moment problem and it may be non-unique.
a(n) ~ 3^(6*n+2) * n^(2*n - 3/2) / (sqrt(Pi) * 2^(2*n+1) * exp(2*n)). - Vaclav Kotesovec, May 24 2025
Showing 1-3 of 3 results.

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