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.

A089156 a(n) = A069722(n+1)^2.

Original entry on oeis.org

0, 16, 576, 25600, 1254400, 65028096, 3497066496, 192980975616, 10855179878400, 619683355033600, 35792910586740736, 2087229562810269696, 122682715414070296576, 7259332273021911040000, 432004345063916175360000, 25835779854133582469529600
Offset: 0

Views

Author

Benoit Cloitre, Jan 03 2004

Keywords

Crossrefs

Cf. A069722.

Programs

  • Mathematica
    Flatten[{0, Table[2^(2*n) * Binomial[2*n, n]^2, {n, 1, 20}]}] (* Vaclav Kotesovec, Sep 28 2019 *)
CoefficientList[Series[-1 + 2*EllipticK[1 - 1/(1 - 64*x)] / (Pi*Sqrt[1 - 64*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 28 2019 *)

Formula

G.f.: 1/AGM(1, (1-64*x)^(1/2)).
E.g.f.: 1 + Sum[n>=0, a(n)*x^(2n)/(2n)! ] = BesselI(0, 4x)^2. - Ralf Stephan, Jan 11 2005
From Vaclav Kotesovec, Sep 28 2019: (Start)
For n > 0, a(n) = 2^(2*n) * binomial(2*n, n)^2.
a(n) ~ 2^(6*n) / (Pi*n). (End)

A069723 a(n) = 2^(n-1)*binomial(2*n-3, n-1).

Original entry on oeis.org

1, 2, 12, 80, 560, 4032, 29568, 219648, 1647360, 12446720, 94595072, 722362368, 5538111488, 42600857600, 328635187200, 2541445447680, 19696202219520, 152935217233920, 1189496134041600, 9265548833587200, 72271280901980160, 564404288948797440, 4412615349963325440
Offset: 1

Views

Author

Valery A. Liskovets, Apr 07 2002

Keywords

Comments

Number of rooted unicursal planar maps with n edges and two vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Links

Crossrefs

Main diagonal of array A082137.
Cf. A069720, A069721, A069722, A082143, A082144, A082145, A088218.

Programs

  • Maple
    Z:=(1-sqrt(1-z))*8^n/sqrt(1-z)/2: Zser:=series(Z, z=0, 33): seq(coeff(Zser, z, n), n=0..20); # Zerinvary Lajos, Jan 16 2007
  • Mathematica
    Table[2^(n - 1) * Binomial[2*n - 3, n - 1], {n, 1,50}] (* G. C. Greubel, Jan 15 2017 *)
  • Sage
    # Assuming offset 0:
A069723  = lambda n: (rising_factorial(n, n)/factorial(n)) << n
[A069723(n) for n in (0..20)] # Peter Luschny, Nov 30 2014

Formula

a(n) = A069722(n)/2, n>1.
G.f.: 4*x/(sqrt(1-8*x) * (1-sqrt(1-8*x))). - Paul Barry, Sep 06 2004
With offset 0: a(n) = (0^n + 2^n*binomial(2n, n))/2. - Paul Barry, Sep 24 2004
D-finite with recurrence (-n+1)*a(n) + 4*(2*n-3)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012
With offset 0: a(n) = 2^n*rf(n,n)/n! = 2^n*A088218(n), where rf denotes the rising factorial. - Peter Luschny, Nov 30 2014
a(n) = Sum_{k=0..n} binomial(n+k-1,k)*binomial(2*n-1, n-k). - Vladimir Kruchinin, Nov 11 2016
a(n) ~ 2^(3*n-4)/sqrt(Pi*n). - Ilya Gutkovskiy, Nov 11 2016
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=1} 1/a(n) = 9/7 + 16*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 7/9 - 8*log(2)/27. (End)

A069721 Number of rooted unicursal planar maps with n edges and no vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Original entry on oeis.org

0, 0, 4, 40, 336, 2688, 21120, 164736, 1281280, 9957376, 77395968, 601968640, 4686094336, 36515020800, 284817162240, 2223764766720, 17379001958400, 135942415319040, 1064286014668800, 8338993950228480, 65388301768458240, 513094808135270400, 4028909667357818880
Offset: 1

Views

Author

Valery A. Liskovets, Apr 07 2002

Keywords

Examples

			G.f. = 4*x^3 + 40*x^4 + 336*x^5 + 2688*x^6 + 21120*x^7 + 164736*x^8 + ...

Links

Crossrefs

Cf. A069720, A069722, A069723.

Programs

  • Magma
    [0] cat [2^(n-2)*(n-2)*Binomial(2*n-2, n-1)/n: n in [2..25]]; // Vincenzo Librandi, Nov 13 2016
  • Maple
    0, seq(2^(n-2)*(n-2)*binomial(2*n-2, n-1)/n, n=2..30); # Robert Israel, Nov 12 2016
  • Mathematica
    a[ n_] := SeriesCoefficient[ ((1 - Sqrt[1 - 8 x])/2)^3 / (2 Sqrt[1 - 8 x] ), {x, 0, n}]; (* Michael Somos, Nov 13 2016 *)

Formula

a(n) = 2^(n-2)*(n-2)*binomial(2n-2, n-1)/n, n>1.
From Robert Israel, Nov 12 2016: (Start)
G.f.: 32*x^3/(sqrt(1-8*x)*(1+sqrt(1-8*x))^3).
E.g.f.: ((1-6*x)/4)*exp(4*x)*I_0(4*x)+(3/2)*exp(4*x)*I_1(4*x)+x/2-1/4, where I_0 and I_1 are modified Bessel functions of the first kind.
a(n+1) = (4*(n-1)*(2*n-1)/((n+1)*(n-2)))*a(n).
a(n) ~ 8^n/(16*sqrt(Pi*n)). (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=3} 1/a(n) = 11/14 - 26*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=3} (-1)^(n+1)/a(n) = 37*log(2)/27 - 13/18. (End)

A335183 T(n,k) = Sum_{j=1..n} 2^j*binomial(2*n-2*j, n-j)*binomial(n+j, n)*binomial(j, k), triangle read by rows (n >= 0 and 0 <= k <= n).

Original entry on oeis.org

0, 4, 4, 36, 60, 24, 288, 688, 560, 160, 2240, 7080, 8760, 5040, 1120, 17304, 68712, 114576, 99456, 44352, 8064, 133672, 642824, 1351840, 1572480, 1055040, 384384, 59136, 1034880, 5864640, 14912064, 21778560, 19536000, 10695168, 3294720, 439296
Offset: 0

Views

Author

Petros Hadjicostas, May 25 2020

Keywords

Comments

This was the original version of A126936.

Examples

			Table T(n,k) (with rows n >= 0 and columns k = 0..n) begins as follows:
       0;
       4,      4;
      36,     60,      24;
     288,    688,     560,     160;
    2240,   7080,    8760,    5040,    1120;
   17304,  68712,  114576,   99456,   44352,   8064;
  133672, 642824, 1351840, 1572480, 1055040, 384384, 59136;
  ...

Crossrefs

Cf. A000984, A067001, A069722 (main diagonal), A126936.

Programs

  • Mathematica
    t[l_, m_] := Sum[2^k*Binomial[2*m-2*k, m-k]*Binomial[m+k, m]*Binomial[k, l], {k, 1, m}]; Table[t[l, m], {m, 0, 7}, {l, 0, m}] // Flatten (* Jean-François Alcover, Jan 09 2014 from the original version of A126936 *)
  • PARI
    T(n,k) = sum(j=1, n, 2^j*binomial(2*n-2*j, n-j)*binomial(n+j, n)*binomial(j, k));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(T(n,k), ", "); ); print(); ); }

Formula

T(n,n) = A069722(n+1) for n >= 0.
T(n,k) = A126936(n,k) = A067001(n,n-k) for n >= k >= 1.
T(n,0) = A126936(n,0) - binomial(2*n, n) = A067001(n,n) - A000984(n) for n >= 0.
Bivariate o.g.f.: Sum_{n,k >= 0} T(n,k)*x^n*y^k = -1/sqrt(1 - 4*x) + sqrt((1 + y)/(1 - 8*x*(1 + y))/(y + sqrt(1 - 8*x*(1 + y)))).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.