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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A079489 a(n) = (2*4^n*binomial(2*n, n) - binomial(4*n + 1, 2*n)) / (n + 1).

Original entry on oeis.org

1, 3, 22, 211, 2306, 27230, 338444, 4362627, 57788170, 781825066, 10757497972, 150073096238, 2117778107732, 30176799215196, 433586825237912, 6274885068167651, 91383942213277530, 1338275570267001458, 19695358741104824036, 291137841642777382330, 4320734864185863437820
Offset: 0

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Author

N. J. A. Sloane, Jan 20 2003

Keywords

Comments

a(n) is the number of ordered trees on 2n-1 edges in which every subtree of the root (including its rooting edge) has an even number of edges, except for the leftmost subtree which has an odd number of edges (including its rooting edge). - David Callan, Apr 10 2012
a(n) is the number of 2 X 2n Young tableaux with a wall between the first and second row in each even column. If there is a wall between two cells, the entries may be decreasing; see [Banderier, Wallner 2021].
Example for a(1)=3:
3 4 2 4 2 3
- - -
1 2, 1 3, 1 4. - Michael Wallner, Mar 09 2022

Links

Crossrefs

Final diagonal of triangle in A078990.
Cf. A000108, A006318, A066357, A060941, A066357, A300386 - A300389, A333563.

Programs

  • Maple
    a := n -> (2*4^n*binomial(2*n, n) - binomial(4*n + 1, 2*n)) / (n + 1):
seq(a(n), n = 0..20);  # Peter Luschny, Aug 26 2024
  • Mathematica
    ((Sqrt[2] Sqrt[1 + Sqrt[1 - 16 x]] - Sqrt[1 - 16 x] - 1)/(4 x) + O[x]^20)[[3]] (* Vladimir Reshetnikov, Sep 25 2016 *)
CoefficientList[Series[-(1 - Sqrt[1 - 4*Sqrt[x]])*(1 - Sqrt[1 + 4*Sqrt[x]])/(4*x), {x,0,50}], x] (* G. C. Greubel, Apr 13 2017 *)
  • PARI
    a(n)=if(n<0,0,polcoeff(serreverse(x*(1-x^2)/(1+x^2)^2+O(x^(2*n+3))),2*n+1))
  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,binomial(4*m-1,2*m)*x^m/m)+x*O(x^n)),n)} \\ Paul D. Hanna, Dec 30 2010

Formula

Series reversion of x(1-x^2)/(1+x^2)^2 expanded in odd powers of x. [Previous name.]
If x = y*(1-y^2)/(1+y^2)^2 then y = x + 3*x^3 + 22*x^5 + 211*x^7 + 2306*x^9 + ...
G.f. A(x) satisfies x*A(x^2) = (C(x) - C(-x))/(C(x) + C(-x)) where C(x) is g.f. of the Catalan numbers A000108.
a(n) = Sum_{k=0..2n} (-1)^k * A000108(2*n-k) * A000108(k). - David Callan, Aug 16 2006
a(n) = ((2^(4n+2))/Gamma(1/2)) * ((Gamma(n+1/2)/(2*Gamma(n+2))) - Gamma(2n+3/2)/Gamma(2n+3)). [David Dickson (dcmd(AT)unimelb.edu.au), Nov 10 2009]
G.f.: exp( Sum_{n>=1} C(4n-1,2n)*x^n/n ). - Paul D. Hanna, Dec 30 2010
G.f.: C(sqrt(x))*C(-sqrt(x)) where C(x) is the g.f. for the Catalan numbers A000108. - David Callan, Apr 10 2012
D-finite with recurrence n*(n+1)*(2*n+1)*a(n) -2*n*(32*n^2-32*n+11)*a(n-1) +16*(4*n-5)*(4*n-3)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 29 2012
a(n) ~ (2-sqrt(2))*16^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 20 2013
a(n) = 2^(2*n+1)*Catalan(n) - Catalan(2*n+1) (see Regev). It follows that the 2-adic valuations of a(n) and Catalan(n) are equal. In particular, a(n) is odd iff n is of the form 2^m - 1. - Peter Bala, Aug 02 2016
G.f.: (sqrt(2) * sqrt(1 + sqrt(1-16*x)) - sqrt(1-16*x) - 1)/(4*x). - Vladimir Reshetnikov, Sep 25 2016
G.f. A(x) satisfies A(x^2) = C(x)^2*r(-x*C(x)^2), where C(x) is g.f. of the Catalan numbers A000108, and r(x) is g.f. of the large Schröder numbers A006318. - Alexander Burstein, Nov 21 2019
From Peter Bala, Sep 14 2021: (Start)
A(x) = exp( Sum_{n >= 1} (1/2)*binomial(4*n,2*n)*x^n/n ).
1 + x*A(x) is the o.g.f. of A066357.
The sequence defined by b(n) := [x^n] A(x)^n begins [1, 3, 53, 1056, 22181, 480003, 10588508, 236720424, ...] and satisfies the congruence b(p) == b(1) (mod p^3) for prime p >= 3. See A333563. Cf. A060941. (End)
From Peter Bala, Oct 23 2024: (Start)
For integer r and positive integer s, define a sequence {u(n) : n >= 0} by setting u(n) = [x^(s*n)] A(x)^(r*n). We conjecture that the supercongruence u(n*p^k) == u(n*p^(k-1)) (mod p^(3*k)) holds for all primes p >= 5 and for all positive integers n and k.
Let B(x) = 1/x * series_reversion(x*A(x)). Define a sequence {v(n) : n >= 0} by setting v(n) = [x^(s*n)] B(x)^(r*n). We conjecture that the supercongruence v(n*p^k) == v(n*p^(k-1)) (mod p^(3*k)) holds for all primes p >= 5 and for all positive integers n and k. (End)

Extensions

New name by Peter Luschny, Aug 26 2024

A333560 Square array read by antidiagonals: T(n,k) = Sum_{j = 0..n*k} binomial(n+j-1,j)*2^j; n,k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 17, 7, 1, 1, 111, 129, 15, 1, 1, 769, 2815, 769, 31, 1, 1, 5503, 65537, 47103, 4097, 63, 1, 1, 40193, 1579007, 3080193, 647167, 20481, 127, 1, 1, 297727, 38862849, 208470015, 109051905, 7929855, 98305, 255, 1, 1, 2228225, 970522623, 14413725697, 19012780031, 3271557121, 90177535, 458753, 511, 1
Offset: 0

Views

Author

Peter Bala, Mar 26 2020

Keywords

Comments

We conjecture that each column sequence satisfies the following supercongruences:
Column k: T(n*p^j, k) == T(n*p^(j-1),k) ( mod p^(3*j) ) for prime p >= 5 and positive integers n and j. Some examples are given below.

Examples

			Square array begins
      |k=0    k=1       k=2           k=3             k=4
  - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  n=0 | 1      1         1             1               1
  n=1 | 1      3         7            15              31
  n=2 | 1     17       129           769            4097
  n=3 | 1    111      2815         47103          647167
  n=4 | 1    769     65537       3080193       109051905
  n=5 | 1   5503   1579007     208470015     19012780031
  n=6 | 1  40193  38862849   14413725697   3385776406529
  n=7 | 1 297727 970522623 1011196362751 611732191969279
  ...
Examples of congruences for column k = 1:
T(5,1) - T(1,1) = 5503 - 3 = (2^2)*(5^3)*11 == 0 ( mod 5^3 ).
T(7,1) - T(1,1) = 297727 - 3 = (2^2)*(7^4)*31 == 0 ( mod 7^3 ).
T(2*11,1) - T(2,1) = 5913649000782757889 - 17 = (2^4)*(3^2)*(11^3)*107*288357478039 == 0 ( mod 11^3 ).
T(5^2,1) - T(5,1) = 2840491845703386005503 - 5503 = (2^7)*(3^3)*(5^6)*7*19*1123*352183001 == 0 ( mod 5^6 ).

Crossrefs

A119259 (column 1), A333561 (column 2), A333562 (column 3). Cf. A333563.

Programs

  • Maple
    T := (n, k) -> add(binomial(n+j-1, j)*2^j, j = 0..n*k):
T_col := k -> seq(T(n, k), n = 0..7):
seq(print(T_col(k)), k = 0..10);

Formula

T(n,k) = Sum_{j = 0..n*k} binomial(n+j-1,j)*2^j.
Conjectural o.g.f. for column k: 2^(k+1)*x*f'(k,(2^k)*x)/(2*f(k,(2^k)*x) - 1) + 1/(1 + x), where f(k,x) = Sum_{n >= 0} 1/((k+1)*n+1)*C((k+1)*n+1,n)* x^n.
Showing 1-2 of 2 results.

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

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