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.

A116637 G.f. satisfies: A(x) = x/series_reversion(x/G(x)) where A(x) + A(-x) = 2*G(x^2) and G(x) is the g.f. of A046646.

Original entry on oeis.org

1, 2, 2, 4, 6, 14, 24, 60, 110, 286, 546, 1456, 2856, 7752, 15504, 42636, 86526, 240350, 493350, 1381380, 2861430, 8064030, 16829280, 47682960, 100134216, 284997384, 601661144, 1719031840, 3645533040, 10450528048, 22249511328, 63967345068
Offset: 0

Views

Author

Paul D. Hanna, Feb 19 2006

Keywords

Examples

			A(x) = 1 + 2*x + 2*x^2 + 4*x^3 + 6*x^4 + 14*x^5 + 24*x^6 + 60*x^7 +...
log(A(x)) = 1*2*x + 2*4/3*x^3 + 7*6/5*x^5 + 30*8/7*x^7 + 143*10/9*x^9 +...

Links

Crossrefs

Cf. A046646, A001764, A006013.

Programs

  • Mathematica
    k := Floor[(n - 1)/2]; Table[If[n == 0, 1, If[Mod[n, 2] == 1, 2*(3*k + 1)!/((k + 1)!*(2*k + 1)!), 2*(3*k + 3)!/((k + 1)!*(2*k + 3)!)]], {n, 0, 50}] (* G. C. Greubel, Nov 21 2017 *)
  • PARI
    {a(n)=local(k=(n-1)\2);if(n==0,1,if(n%2==1, 2*(3*k+1)!/((k+1)!*(2*k+1)!), 2*(3*k+3)!/((k+1)!*(2*k+3)!)))}
for(n=0,40,print1(a(n),", "))
  • PARI
    {a(n)=if(n<1, n==0, 2*(n+n\2)!/ (n\2+n%2)!/ (n+1-(n%2))!)} /* Michael Somos, Feb 22 2006 */

Formula

a(2*n+1) = 2*(3*n+1)!/((n+1)!*(2*n+1)!) = 2*A006013(n), with a(0)=1 and a(2*n+2) = 2*(3*n+3)!/((n+1)!*(2*n+3)!) = 2*A001764(n+1).
G.f. satisfies: A(x) = G(x/A(x)) and A(x*G(x)) = G(x), where G(x) is the g.f. of A046646.
G.f. satisfies: A(x) = 1/A(-x) since log(A(x)) = Sum_{n>=0} 2*A006013(n)*(n+1)/(2n+1)*x^(2n+1) is an odd function.
G.f.: (1+v)/(1-v) where v=2*sqrt(3)*sin(asin(3*sqrt(3)*x/2)/3)/3. - Paul Barry, Jul 07 2007
Conjecture: 4*n*(n+1)*(3*n-1)*a(n) -36*n*a(n-1) -3*(3*n-5)*(3*n+2)*(3*n-4)*a(n-2)=0. - R. J. Mathar, Jun 22 2016

A091599 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with r*n edges and a fixed outer face of r*k edges which are invariant under a rotation of 1/r for any r >= 2 (independent of actual value of r).

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 24, 26, 12, 1, 110, 120, 75, 20, 1, 546, 594, 416, 174, 30, 1, 2856, 3094, 2289, 1176, 350, 42, 1, 15504, 16728, 12768, 7322, 2880, 636, 56, 1, 86526, 93024, 72420, 44388, 20475, 6324, 1071, 72, 1, 493350, 528770, 417240, 267240, 136252, 51495, 12740, 1700, 90, 1
Offset: 1

Views

Author

Emeric Deutsch, Mar 03 2004

Keywords

Comments

Table I in the Brown reference.

Examples

			Triangle starts:
    1;
    2,   1;
    6,   6,  1;
   24,  26, 12,  1;
  110, 120, 75, 20, 1;
  ...

Links

Crossrefs

Column 1 gives A046646, column 2 gives A046647, row sums give A000259.
Same as A046651 but with rows reversed.
Cf. A046653, A091665.

Programs

  • Maple
    T := proc(n,k) if k<=n then k*sum((2*j-k)*(j-1)!*(3*n-j-k-1)!/(j-k)!/(j-k)!/(2*k-j)!/(n-j)!,j=k..min(n,2*k))/(2*n-k)! else 0 fi end: seq(seq(T(n,k), k=1..n),n=1..11);
  • PARI
    T(n, k) = k*sum(j=k, min(n, 2*k), (2*j-k)*(j-1)!*(3*n-j-k-1)!/(((j-k)!)^2*(2*k-j)!*(n-j)!))/(2*n-k)!
for(n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print) \\ Andrew Howroyd, Mar 29 2021

Formula

T(n, k) = k*(Sum_{j=k..min(n, 2*k)} (2*j-k)*(j-1)!*(3*n-j-k-1)!/(((j-k)!)^2*(2*k-j)!*(n-j)!))/(2*n-k)!

Extensions

Name clarified by Andrew Howroyd, Mar 29 2021

A046647 Number of certain rooted planar maps.

Original entry on oeis.org

1, 6, 26, 120, 594, 3094, 16728, 93024, 528770, 3058770, 17948970, 106585440, 639318456, 3867821640, 23574446992, 144621823632, 892293152994, 5533289372170, 34468829508750, 215594574231960, 1353464311979010
Offset: 2

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

A diagonal of A046651. Cf. A046646.

Formula

a(2)=1; a(n) = 4*(7*n-15)*(3*n-6)!/((n-2)!*(2n-2)!). - Emeric Deutsch, Mar 03 2004
G.f.: (g+1)*(3*g+1)/(g-1)^2 where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011

Extensions

More terms from Emeric Deutsch, Mar 03 2004

A101401 Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges in which the leftmost child of the root has degree k.

Original entry on oeis.org

1, 1, 2, 3, 6, 3, 12, 24, 15, 4, 55, 110, 75, 28, 5, 273, 546, 390, 168, 45, 6, 1428, 2856, 2100, 980, 315, 66, 7, 7752, 15504, 11628, 5712, 2040, 528, 91, 8, 43263, 86526, 65835, 33516, 12825, 3762, 819, 120, 9, 246675, 493350, 379500, 198352, 79695, 25410, 6370, 1200, 153, 10
Offset: 1

Views

Author

Emeric Deutsch, Jan 15 2005

Keywords

Comments

Row n contains n terms. Column k=0 and row sums yield the ternary numbers (A001764).

Examples

			T(2,0)=1 and T(2,1)=2 because the noncrossing trees with 2 edges are /\, |_ and _|.
Triangle starts:
    1;
    1,   2;
    3,   6,   3;
   12,  24,  15,  4;
   55, 110,  75,  28,  5;
  273, 546, 390, 168, 45, 6;
  ...

Links

Crossrefs

Cf. A001764.
Column k=1 is A046646.

Programs

  • Maple
    T:=proc(n,k) if n=1 and k=1 then 0 elif k<=n then (k+1)*(2*k+1)*binomial(3*n-k-2,2*n-1)/(3*n-k-2) else 0 fi end: for n from 1 to 10 do seq(T(n,k),k=0..n-1) od; # yields sequence in triangular form
  • Mathematica
    T[n_, k_] := ((k + 1)(2k + 1)/(3n - k - 2)) Binomial[3n - k - 2, 2n - 1];
Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Apr 10 2020 *)
  • PARI
    T(n, k) = if(kAndrew Howroyd, Nov 06 2017

Formula

T(n, k) = ((k+1)(2k+1)/(3n-k-2)) binomial(3n-k-2, 2n-1).
G.f.: zg/(1-tzg^2)^2, where g = 1+zg^3 is the g.f. of the ternary numbers (A001764).
Showing 1-4 of 4 results.

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

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