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.

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A049171 Revert transform of 2*x*(1-x)-x/(1+x).

Original entry on oeis.org

1, 1, 3, 9, 33, 125, 503, 2081, 8849, 38345, 168875, 753401, 3398177, 15469493, 70984559, 327982529, 1524644897, 7125440913, 33459931155, 157794990633, 747021246817, 3548843286829, 16912921740775, 80836929471329, 387397148131889, 1861088017162457
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Sign diagram of generating sequence: +++-------------...
The sequences A049171 to A049189 are defined by series reversion of a sequence with rational (ordinary) generating function g(x). Solving g(x)=y for x yields algebraic equations for x, so the sequences have P-finite recurrences. - R. J. Mathar, Jul 24 2023

Links

Programs

  • Mathematica
    Table[Sum[Binomial[3*k,k]*Binomial[n-1+k,3*k]/(2k+1)*2^k,{k,0,Floor[(n-1)/2]}],{n,1,20}] (* Vaclav Kotesovec, Oct 24 2012 *)

Formula

a(n+1) = Sum_{k=0..floor(n/2)} A108759(n,k)*2^k. - Philippe Deléham, Dec 08 2009
Recurrence: 4*(n-1)*n*a(n) = 2*(n-1)*(5*n-6)*a(n-1) + 3*(16*n^2 - 67*n + 69)*a(n-2) + (25*n^2 - 169*n + 285)*a(n-3) + (n-4)*(2*n-9)*a(n-4). - Vaclav Kotesovec, Oct 24 2012
a(n) ~ sqrt(sqrt(3)-1)*((5+3*sqrt(3))/2)^n/(2*sqrt(6*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012

Extensions

NAME corrected by R. J. Mathar, Jul 23 2023

A364757 The pyramidal array T(r,g,b) = (r+g+b)/((g+b)*(r+b))*C(r+g,b-1)*C(g+b,r)*C(r+b,g), where 1 <= b <= ceiling((r+g+b)/2) and 0 <= r,g <= floor((r+g+b)/2). Read first over the layers corresponding to fixed sum r+g+b, then over the diagonals corresponding to fixed b.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 2, 2, 1, 8, 1, 5, 15, 15, 1, 5, 1, 3, 3, 8, 54, 8, 1, 27, 27, 1, 7, 70, 70, 42, 168, 42, 1, 14, 14, 1, 4, 4, 30, 192, 30, 20, 400, 400, 20, 1, 64, 200, 64, 1, 9, 210, 210, 405, 1500, 405, 90, 900, 900, 90, 1, 30, 81, 30, 1, 5, 5, 80, 500, 80, 147, 2625, 2625, 147, 40, 1750, 5000, 1750, 40, 1, 125, 875, 875, 125, 1
Offset: 1

Views

Author

Robert Muth, Aug 05 2023

Keywords

Comments

T(r,g,b) is the number of injectively 3-colored trees with r red vertices, g green vertices, and b blue vertices, including a root vertex which is colored blue.
Summing T(r,g,b) over all r,g,b such that r+g+b=n yields the n-th Catalan number, A000108(n).
Column (or row) sums within each fixed r+g+b=n layer yield the number of ordered trees on n edges containing a fixed number of nodes adjacent to a leaf, A108759(n).
Main antidiagonal (corresponding to maximal value b = ceiling((r+g+b)/2)) within each fixed odd (r+g+b) layer is the number of "fighting fish" with fixed numbers of left lower free and right lower free edges with a marked tail A278880.

Examples

			The first few layers of the pyramidal array are:
-----------------------------------------------------------------------
      1           (r+g+b=1), (b=1)           T(0,0,1)
                                                         LAYER SUM:   1
-----------------------------------------------------------------------
     1 1          (r+g+b=2), (b=1)        T(0,1,1) T(1,0,1)
                                                         LAYER SUM:   2
-----------------------------------------------------------------------
      3           (r+g+b=3), (b=1)           T(1,1,1)
     1 1          (r+g+b=3), (b=2)        T(0,1,2) T(1,0,2)
                                                         LAYER SUM:   5
-----------------------------------------------------------------------
     2 2          (r+g+b=4), (b=1)        T(1,2,1) T(2,1,1)
    1 8 1         (r+g+b=4), (b=2)     T(0,2,2) T(1,1,2) T(2,0,2)
                                                         LAYER SUM:  14
-----------------------------------------------------------------------
      5           (r+g+b=5), (b=1)           T(2,2,1)
    15 15         (r+g+b=5), (b=2)        T(1,2,2) T(2,1,2)
   1  5  1        (r+g+b=5), (b=3)     T(0,2,3) T(1,1,3) T(2,0,3)
                                                         LAYER SUM:  42
-----------------------------------------------------------------------
     3  3         (r+g+b=6), (b=1)        T(2,3,1) T(3,2,1)
   8  54  8       (r+g+b=6), (b=2)     T(1,3,2) T(2,2,2) T(3,1,2)
 1  27  27  1     (r+g+b=6), (b=3)  T(0,3,3) T(1,2,3) T(2,1,3) T(3,0,3)
                                                         LAYER SUM: 132
-----------------------------------------------------------------------
      7           (r+g+b=7), (b=1)            T(3,3,1)
   70   70        (r+g+b=7), (b=2)        T(2,3,2) T(3,2,2)
 42  168  42      (r+g+b=7), (b=3)     T(1,3,3) T(2,2,3) T(3,1,3)
1   14  14   1    (r+g+b=7), (b=4)  T(0,3,4) T(1,2,4) T(2,1,4) T(3,0,4)
                                                         LAYER SUM: 429
-----------------------------------------------------------------------

Links

Crossrefs

Cf. A000108, A108759, A278880.

Formula

T(r,g,b) = ((r+g+b)/((g+b)*(r+b)))*C(r+g,b-1)*C(g+b,r)*C(r+b,g).
T(r,g,b) = ((r+g+b)/((g+b)*(r+b))) * ((r+g)!/((r+g-b+1)!*(b-1)!)) * ((g+b)!/((g+b-r)!*r!)) * ((r+b)!/((r+b-g)!*g!)).
Showing 1-2 of 2 results.

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