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.

A052510 Number of labeled planar binary trees with 2n-1 elements (external nodes or internal nodes).

Original entry on oeis.org

1, 6, 240, 25200, 5080320, 1676505600, 821966745600, 560992303872000, 508633022177280000, 591438478187741184000, 858123464716031754240000, 1519736656012092236759040000, 3226517823533365056503808000000, 8089341114715793820234547200000000
Offset: 1

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Equals central terms of A174449. - G. C. Greubel, Nov 29 2021

Links

Crossrefs

Cf. A000108, A036770, A101921, A174449.

Programs

  • Magma
    [Factorial(2*n-1)*Catalan(n-1): n in [1..15]]; // G. C. Greubel, Nov 29 2021
  • Maple
    spec := [S, {S=Union(Z, Prod(Z, S, S))}, labeled]:
seq(combstruct[count](spec, size=2*n-1), n=1..14);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, n,
      4*(n-1)*(2*n-3)*(2*n-1)*a(n-1)/n)
    end:
seq(a(n), n=1..22);  # Alois P. Heinz, Dec 03 2019
  • Mathematica
    nn=20;f[x_]:=Sum[a[n]x^n/n!,{n,0,nn}];s=SolveAlways[0==Series[f[x]-x(1+f[x]^2),{x,0,nn}],x];Select[Flatten[Table[a[n],{n,0,nn}]/.s],#>0&]  (* Geoffrey Critzer, Mar 23 2013 *)
RecurrenceTable[{a[1]==1, n*a[n]==(4*(n-1)*(2*n-3)*(2*n-1))*a[n-1]}, a[n], {n,1,22}] (* Georg Fischer, Dec 03 2019 following Alois P. Heinz *)
a[n_]:= CatalanNumber[n-1] Gamma[2n]; Array[a,14] (* Peter Luschny, Dec 03 2019 *)
  • PARI
    a=vector(28);print1(a[1]=1,", ");forstep(k=1,#a-2,2,print1(a[k+2]=4*a[k]*(k^3+3*k^2+2*k)/(k+3),", ")) \\ Hugo Pfoertner, Dec 04 2019
  • Sage
    [factorial(2*n-1)*catalan_number(n-1) for n in (1..15)] # G. C. Greubel, Nov 29 2021

Formula

E.g.f.: ((1/2)/x)*(1-sqrt(1-4*x^2)). [With interspersed zeros.]
Recurrence: b(1)=1, b(2)=0, b(n)=(4*n^3-12*n^2+8*n)*b(n-2)/(n+1) and a(n) = b(2*n-1).
a(n) = (2n-1)/n * ( (2(n-1))! / (n-1)! )^2. - Travis Kowalski (kowalski(AT)euclid.UCSD.Edu), Dec 15 2000
i*sin(arcsec(2*x)) = -1/(2*x) + x + 6*x^3/3! + 240*x^5/5! + 25200*x^7/7! + ...
a(n) = 2^(n-1) * A036770(n).
a(n) = (2*n-1)! * A000108(n-1). - Michail Stamatakis, Jan 24 2019
Sum_{n>=1} 1/a(n) = 1 + StruveL(0, 1/2)*Pi/8 + StruveL(1, 1/2)*Pi/4, where StruveL is the modified Struve function. - Amiram Eldar, Dec 04 2022

Extensions

Edited by Georg Fischer, Dec 03 2019

A174450 Triangle T(n, k, q) = n!*(n+1)!*q^k/((n-k)!(n-k+1)!) if floor(n/2) > k-1, otherwise n!*(n+1)!*q^(n-k)/(k!*(k+1)!) for q = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 24, 24, 1, 1, 40, 960, 40, 1, 1, 60, 2400, 2400, 60, 1, 1, 84, 5040, 201600, 5040, 84, 1, 1, 112, 9408, 564480, 564480, 9408, 112, 1, 1, 144, 16128, 1354752, 81285120, 1354752, 16128, 144, 1, 1, 180, 25920, 2903040, 243855360, 243855360, 2903040, 25920, 180, 1
Offset: 0

Views

Author

Roger L. Bagula, Mar 20 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,   1;
  1,  12,     1;
  1,  24,    24,       1;
  1,  40,   960,      40,         1;
  1,  60,  2400,    2400,        60,         1;
  1,  84,  5040,  201600,      5040,        84,       1;
  1, 112,  9408,  564480,    564480,      9408,     112,     1;
  1, 144, 16128, 1354752,  81285120,   1354752,   16128,   144,   1;
  1, 180, 25920, 2903040, 243855360, 243855360, 2903040, 25920, 180,  1;

Links

Crossrefs

Cf. A174449 (q=1), this sequence (q=2), A174451 (q=3).
Cf. A000108.

Programs

  • Magma
    F:= Factorial; // T = A174450
T:= func< n,k,q | Floor(n/2) gt k-1 select F(n)*F(n+1)*q^k/(F(n-k)*F(n-k+1)) else F(n)*F(n+1)*q^(n-k)/(F(k)*F(k+1)) >;
[T(n,k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 29 2021
  • Mathematica
    T[n_, k_, q_]:= If[Floor[n/2]>k-1, n!*(n+1)!*q^k/((n-k)!*(n-k+1)!), n!*(n+1)!*q^(n-k)/(k!*(k+1)!)];
Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten
  • Sage
    f=factorial
def A174450(n,k,q):
    if ((n//2)>k-1): return f(n)*f(n+1)*q^k/(f(n-k)*f(n-k+1))
    else: return f(n)*f(n+1)*q^(n-k)/(f(k)*f(k+1))
flatten([[A174450(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 29 2021

Formula

T(n, k, q) = n!*(n+1)!*q^k/((n-k)!(n-k+1)!) if floor(n/2) > k-1, otherwise n!*(n+1)!*q^(n-k)/(k!*(k+1)!) for q = 2.
T(n, n-k, q) = T(n, k, q).
From G. C. Greubel, Nov 29 2021: (Start)
T(2*n, n, q) = q^n*(2*n+1)!*Catalan(n) for q = 2.
T(n, k, q) = binomial(n, k)*binomial(n+1, k+1) * ( k!*(k+1)!*q^k/(n-k+1) if (floor(n/2) >= k), otherwise ((n-k)!)^2*q^(n-k) ), for q = 2. (End)

Extensions

Edited by G. C. Greubel, Nov 29 2021

A174451 Triangle T(n, k, q) = n!*(n+1)!*q^k/((n-k)!(n-k+1)!) if floor(n/2) > k-1, otherwise n!*(n+1)!*q^(n-k)/(k!*(k+1)!) for q = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 18, 1, 1, 36, 36, 1, 1, 60, 2160, 60, 1, 1, 90, 5400, 5400, 90, 1, 1, 126, 11340, 680400, 11340, 126, 1, 1, 168, 21168, 1905120, 1905120, 21168, 168, 1, 1, 216, 36288, 4572288, 411505920, 4572288, 36288, 216, 1, 1, 270, 58320, 9797760, 1234517760, 1234517760, 9797760, 58320, 270, 1
Offset: 0

Views

Author

Roger L. Bagula, Mar 20 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,   1;
  1,  18,     1;
  1,  36,    36,       1;
  1,  60,  2160,      60,          1;
  1,  90,  5400,    5400,         90,          1;
  1, 126, 11340,  680400,      11340,        126,       1;
  1, 168, 21168, 1905120,    1905120,      21168,     168,     1;
  1, 216, 36288, 4572288,  411505920,    4572288,   36288,   216,   1;
  1, 270, 58320, 9797760, 1234517760, 1234517760, 9797760, 58320, 270, 1;

Links

Crossrefs

Cf. A174449 (q=1), A174450 (q=2), this sequence (q=3).
Cf. A000108.

Programs

  • Magma
    F:= Factorial; // T = A174451
T:= func< n,k,q | Floor(n/2) gt k-1 select F(n)*F(n+1)*q^k/(F(n-k)*F(n-k+1)) else F(n)*F(n+1)*q^(n-k)/(F(k)*F(k+1)) >;
[T(n,k,3): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 29 2021
  • Mathematica
    T[n_, k_, q_]:= If[Floor[n/2]>k-1, n!*(n+1)!*q^k/((n-k)!*(n-k+1)!), n!*(n+1)!*q^(n-k)/(k!*(k+1)!)];
Table[T[n, k, 3], {n,0,12}, {k,0,n}]//Flatten
  • Sage
    f=factorial
def A174451(n,k,q):
    if ((n//2)>k-1): return f(n)*f(n+1)*q^k/(f(n-k)*f(n-k+1))
    else: return f(n)*f(n+1)*q^(n-k)/(f(k)*f(k+1))
flatten([[A174451(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 29 2021

Formula

T(n, k, q) = n!*(n+1)!*q^k/((n-k)!(n-k+1)!) if floor(n/2) > k-1, otherwise n!*(n+1)!*q^(n-k)/(k!*(k+1)!) for q = 3.
T(n, n-k, q) = T(n, k, q).
From G. C. Greubel, Nov 29 2021: (Start)
T(2*n, n, q) = q^n*(2*n+1)!*Catalan(n) for q = 3.
T(n, k, q) = binomial(n, k)*binomial(n+1, k+1) * ( k!*(k+1)!*q^k/(n-k+1) if (floor(n/2) >= k), otherwise ((n-k)!)^2*q^(n-k) ), for q = 3. (End)

Extensions

Edited by G. C. Greubel, Nov 29 2021
Showing 1-3 of 3 results.

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