A378061 -id:A378061 - cp's OEIS frontend

A018224 a(n) = binomial(n, floor(n/2))^2 = A001405(n)^2.

1, 1, 4, 9, 36, 100, 400, 1225, 4900, 15876, 63504, 213444, 853776, 2944656, 11778624, 41409225, 165636900, 590976100, 2363904400, 8533694884, 34134779536, 124408576656, 497634306624, 1828114918084, 7312459672336, 27043120090000, 108172480360000, 402335398890000

Offset: 0

N. J. A. Sloane

a(n) is also the number of rooted two-vertex (or, dually, two-face) regular planar maps of valency n+1. - Valery A. Liskovets, Oct 19 2005
If A is a random matrix in USp(4) (4 X 4 complex matrices that are unitary and symplectic), then a(n)=(-1)^n*E[(tr(A^4))^n]. - Andrew V. Sutherland, Apr 01 2008
Number of square lattice walks with unit steps in all four directions (NSWE), starting at the origin, ending on the y-axis, and never going below the x-axis. Row sums of A378061. - Peter Luschny, Dec 08 2024

			The 9 lattice walks defined in the comments: 'NNN', 'NNS', 'NSN', 'NWE', 'NEW', 'WNE', 'WEN', 'ENW', 'EWN'.

Stephanie Anderson, Table for n, a(n) for n = 0..200
Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Slides, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Synthesis, Séminaire de Combinatoire Ph. Flajolet, June 06 2013.
Alin Bostan, Calcul Formel pour la Combinatoire des Marches [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d'Informatique de Paris Nord, Université Paris 13, December 2017.
Alin Bostan, Frédéric Chyzak, Mark van Hoeij, Manuel Kauers, and Lucien Pech, Hypergeometric expressions for generating functions of walks with small steps in the quarter plane, Eur. J. Comb. 61, 242-275 (2017).
M. Bousquet, G. Labelle and P. Leroux, Enumeration of planar two-face maps, Discrete Math., vol. 222 (2000), 1-25.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Helmut Prodinger, Two New Identities Involving the Catalan Numbers: A classical approach, arXiv:1911.07604 [math.CO], 2019.

Bisections are A002894 and A060150.

Cf. A000108, A001405, A056040, A113182, A138545, A378061.

s := x -> (1+x)*EllipticK(x)/(x*Pi/2)-1/x:
seq(4^n*coeff(series(s(x),x,n+2),x,n),n=0..23); # Peter Luschny, Oct 14 2015

(* Note that Mathematica uses a different definition of the EllipticK function. *)
CoefficientList[Series[(-Pi + (2 + 8 x) EllipticK[16 x^2])/(4 Pi x), {x,0,23}], x] (* Peter Luschny, Oct 14 2015 *)
Table[Binomial[n,Floor[n/2]]^2,{n,0,30}] (* Harvey P. Dale, Dec 02 2022 *)

vector(50, n, n--; binomial(n, n\2)^2) \\ Altug Alkan, Oct 14 2015

E.g.f.: BesselI(0, 2*x)*(BesselI(0, 2*x)+BesselI(1, 2*x)). - Vladeta Jovovic, Jun 12 2005
G.f. (1+1/(4*x))*hypergeom([1/2, 1/2],[1],16*x^2)-1/(4*x). - Mark van Hoeij, Oct 13 2009
a(n) = (n!/(floor(n/2)!*floor((n+1)/2)!))^2. - Peter Luschny, Apr 29 2014
a(n) = A056040(n) * A056040(n+1) / (n+1). - Peter Luschny, Apr 29 2014
a(n) = 4^n*[x^n]((1+x)*EllipticK(x)/(x*Pi/2)-1/x). - Peter Luschny, Oct 14 2015
a(n) ~ 4^n*((2*n+3)/(2*n+1))^((-1)^n/2)/((n+1)*Pi/2). - Peter Luschny, Oct 14 2015
a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*C(k)*binomial(2*n-2*k,n-k) where C(k) are Catalan numbers (A000108), see Prodinger. - Michel Marcus, Nov 19 2019
From Peter Bala, Jul 03 2023: (Start)
Right hand side of the binomial sum identity (1/2)*Sum_{k = 0..n+1} (-1)^k*4^(n+1-k)*binomial(n+1,k)*binomial(n+k,k)*binomial(2*k,k) = a(n).
a(n) = (1/2)*4^(n+1) * hypergeom([n+1, -n-1, 1/2], [1, 1], 1).
P-recursive:
(2*n - 1)*(n + 1)^2*a(n) = 4*(2*n^2 - 1)*a(n-1) + 16*(2*n + 1)*(n - 1)^2*a(n-2) with a(0) = a(1) = 1. (End)

A378060 a(n) = binomial(n, floor((n-1)/2))^2.

Original entry on oeis.org

0, 1, 1, 9, 16, 100, 225, 1225, 3136, 15876, 44100, 213444, 627264, 2944656, 9018009, 41409225, 130873600, 590976100, 1914762564, 8533694884, 28210561600, 124408576656, 418151049316, 1828114918084, 6230734868736, 27043120090000, 93271169290000, 402335398890000

Offset: 0

Peter Luschny, Dec 03 2024

Number of walks of length n with unit steps in all four directions (NSWE), starting at the origin and ending on the y-axis, never going below the x-axis and the end point having a positive height.

			The 16 walks of length 4: NNNN, NNNS, NNSN, NNEW, NNWE, NSNN, NENW, NEWN, NWNE, NWEN, ENNW, ENWN, EWNN, WNNE, WNEN, WENN.

Cf. A060150 (odd bisection), A337900 (even bisection), A037952, A378061.

# Generates the walks (for illustration only).
function aCount(n::Int)
    a = [""]
    c = 0
    for w in a
        if length(w) == n
            if (count('N', w) != count('S', w) && count('W', w) == count('E', w))
                c += 1
                # println(w)
            end
        else
            for j in "NSEW"
                u = string(w, j)
                if count('N', u) >= count('S', u)
                   push!(a, u)
    end end end end
    return c
end
println([aCount(n) for n in 0:11])

a := n -> binomial(n, iquo(n+1, 2) - 1)^2: seq(a(n), n = 0..27);
a := proc(n) option remember; if n < 2 then n else ((32*n^5 - 48*n^4 + 16*n^2)*a(n - 2) + (8*n^4 - 20*n^2)*a(n - 1))/(2*(n - 1)^2*(n - 1/2)*(n + 2)^2) fi end:
# Alternative:
egf := BesselI(0, 2*x)^2 + BesselI(1, 2*x)*BesselI(0, 2*x)*(1 - 1/x):
ser := series(egf, x, 29): seq(n!*coeff(ser, x, n), n = 0..27);

Array[Binomial[#, Floor[(# + 1)/2] - 1]^2 &, 28, 0] (* Michael De Vlieger, Dec 04 2024 *)

a(n) = n!*[x^n] (BesselI(0, 2*x)^2 + BesselI(1, 2*x)*BesselI(0, 2*x)*(1 - 1/x)).
a(n) = [x^n] (((8*x^2 + 2*x)*EllipticK(4*x) - Pi*(1 + x) + 2*EllipticE(4*x))/(4*x^2*Pi)).
a(n) = [x^n] (x*hypergeom([1,3/2,3/2], [2,2], 16*x^2) + x^2*hypergeom([3/2,3/2,2,2], [1,3,3], 16*x^2)).
a(n) = Sum_{k=0..n} (-1)^(n-k+N)*C(n-k, N)*C(n, k)*C(n+k, k), where N = floor((n-1)/2) and C = binomial.
Recurrence: a(n) = ((32*n^5 - 48*n^4 + 16*n^2)*a(n - 2) + (8*n^4 - 20*n^2)*a(n - 1))/(2*(n - 1)^2*(n - 1/2)*(n + 2)^2).
a(n) = Sum_{k=1..n} A378061(n, k).

A378062 Array read by ascending antidiagonals: A(n, k) = (n + 1)binomial(2k + n - 1, k - 1)^2 / (2*k + n - 1) for k > 0, and A(n, 0) = 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 8, 20, 0, 1, 15, 75, 175, 0, 1, 24, 189, 784, 1764, 0, 1, 35, 392, 2352, 8820, 19404, 0, 1, 48, 720, 5760, 29700, 104544, 226512, 0, 1, 63, 1215, 12375, 81675, 382239, 1288287, 2760615, 0, 1, 80, 1925, 24200, 196625, 1145144, 5010005, 16359200, 34763300

Offset: 0

Peter Luschny, Dec 07 2024

			Array A(n, k) starts:
  [0] 0, 1,  3,   20,   175,    1764,    19404, ... A000891
  [1] 0, 1,  8,   75,   784,    8820,   104544, ... A145600
  [2] 0, 1, 15,  189,  2352,   29700,   382239, ... A145601
  [3] 0, 1, 24,  392,  5760,   81675,  1145144, ... A145602
  [4] 0, 1, 35,  720, 12375,  196625,  3006003, ... A145603
  [5] 0, 1, 48, 1215, 24200,  429429,  7154784, ...
  [6] 0, 1, 63, 1925, 44044,  869505, 15767024, ...
  [7] 0, 1, 80, 2904, 75712, 1656200, 32626944, ...
.
Seen as a triangle, T(n, k) = A(n-k, k). Compare the descending antidiagonals of A378061.
  [0] 0;
  [1] 0, 1;
  [2] 0, 1,  3;
  [3] 0, 1,  8,  20;
  [4] 0, 1, 15,  75,  175;
  [5] 0, 1, 24, 189,  784,  1764;
  [6] 0, 1, 35, 392, 2352,  8820,  19404;
  [7] 0, 1, 48, 720, 5760, 29700, 104544, 226512;

Rows: A000891, A145600, A145601, A145602, A145603.

Columns: A005563, A005565, A378061.

A := (n, k) -> ifelse(k = 0, 0, (n + 1)*binomial(2*k + n - 1, k - 1)^2/(2*k + n - 1)):
for n from 0 to 7 do seq(A(n, k), k = 0..7);

A[n_, k_] := If[k==0, 0, (n + 1)*Binomial[2*k + n - 1, k - 1]^2 / (2*k + n - 1)]; Table[A[n-k,k],{n,0,9},{k,0,n}]//Flatten (* Stefano Spezia, Dec 08 2024 *)

cp's OEIS Frontend

A018224 a(n) = binomial(n, floor(n/2))^2 = A001405(n)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A378060 a(n) = binomial(n, floor((n-1)/2))^2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Julia

Maple

Mathematica

Formula

A378062 Array read by ascending antidiagonals: A(n, k) = (n + 1)binomial(2k + n - 1, k - 1)^2 / (2*k + n - 1) for k > 0, and A(n, 0) = 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

cp's OEIS Frontend

A018224 a(n) = binomial(n, floor(n/2))^2 = A001405(n)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A378060 a(n) = binomial(n, floor((n-1)/2))^2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Julia

Maple

Mathematica

Formula

A378062 Array read by ascending antidiagonals: A(n, k) = (n + 1)*binomial(2*k + n - 1, k - 1)^2 / (2*k + n - 1) for k > 0, and A(n, 0) = 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

A378062 Array read by ascending antidiagonals: A(n, k) = (n + 1)binomial(2k + n - 1, k - 1)^2 / (2*k + n - 1) for k > 0, and A(n, 0) = 0.