A000888 -id:A000888 - cp's OEIS frontend

A151392 Duplicate of A000888.

1, 2, 12, 100, 980, 10584, 121968, 1472328, 18404100, 236390440, 3103161776, 41469525552, 562496897872, 7726605740000, 107289439704000, 1503840313184400, 21252802073091300

Offset: 0

dead

A013709 a(n) = 4^(2*n+1).

Original entry on oeis.org

4, 64, 1024, 16384, 262144, 4194304, 67108864, 1073741824, 17179869184, 274877906944, 4398046511104, 70368744177664, 1125899906842624, 18014398509481984, 288230376151711744, 4611686018427387904, 73786976294838206464, 1180591620717411303424, 18889465931478580854784

Offset: 0

N. J. A. Sloane

Also powers of 2 with singly even numbers (A016825) as exponents. - Alonso del Arte, Sep 03 2012

The partial sum of A000888(n) = Catalan(n)^2*(n + 1) resp. A267844(n) = Catalan(n)^2*(4n + 3) resp. A267987(n) = Catalan(n)^2*(4n + 4) divided by A013709(n) (this) a(n) = 2^(4n+2) absolutely converge to 1/Pi resp. 1 resp. 4/Pi. Thus this series is 1/Pi resp. 1 resp. 4/Pi. - Ralf Steiner, Jan 23 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (16).

Cf. A000888, A001025, A016825, A267844, A267987.

Cf. A000302, A005408.

[4^(2*n+1): n in [0..20]]; // Vincenzo Librandi, May 26 2011

A013709:=n->4^(2*n+1): seq(A013709(n), n=0..20); # Wesley Ivan Hurt, Jan 30 2016

2^(4 Range[0, 15] + 2) (* Alonso del Arte, Sep 03 2012 *)
NestList[16#&,4,20] (* Harvey P. Dale, Jun 03 2013 *)

a(n)=4<<(4*n) \\ Charles R Greathouse IV, Apr 07 2012

a(n) = 16*a(n-1), n > 0; a(0) = 4. G.f.: 4/(1 - 16*x). [Philippe Deléham, Nov 23 2008]
a(n) = 4^(2*n + 1) = 2^(4*n + 2). - Alonso del Arte, Sep 03 2012
a(n) = 4*A001025(n). - Michel Marcus, Jan 30 2016
From Elmo R. Oliveira, Aug 26 2024: (Start)
E.g.f.: 4*exp(16*x).
a(n) = A000302(A005408(n)). (End)

A005558 a(n) is the number of n-step walks on square lattice such that 0 <= y <= x at each step.

Original entry on oeis.org

1, 1, 3, 6, 20, 50, 175, 490, 1764, 5292, 19404, 60984, 226512, 736164, 2760615, 9202050, 34763300, 118195220, 449141836, 1551580888, 5924217936, 20734762776, 79483257308, 281248448936, 1081724803600, 3863302870000, 14901311070000, 53644719852000

Offset: 0

N. J. A. Sloane

Number of n-step walks that start at the origin, constrained to stay in the first octant (0 <= y <= x). (Conjectured) - Benjamin Phillabaum, Mar 11 2011, corrected by Robert Israel, Oct 07 2015
For n >= 1, a(n-1) is the number of Dyck Paths with semilength n having floor((n+2)/2) U's in odd numbered positions. Example: (U is in odd numbered position and u is in even numbered position) Dyck path with n=5, floor ((5+2)/2)=3: UuddUuUddd. - Roger Ford, May 27 2017
The ratio of the number of n-step walks on the octant with an equal number of North steps and South steps to the total number of n-step walks on the octant is A005817(n)/a(n).  For the reduced ratio, if n is divisible by 4 or n-1 is divisible by 4 the ratio is 1:floor(n/4)+1 and for all other values of n the ratio is 2:floor(n/2)+2.  Example n = 4: A005817(4) = 10; EEEE, EEEW, EEWE, EWEE, EWEW, EEWW, ENSE, ENES, ENSW, EENS; a(4) = 20; 10:20 reduces to 1:2. - Roger Ford, Nov 04 2019

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Slides for Séminaire de Combinatoire Ph. Flajolet, Mar 28 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)
Sergi Elizalde, The degree of symmetry of lattice paths, arXiv:2002.12874 [math.CO], 2020.
R. K. Guy, Letter to N. J. A. Sloane, May 1990
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6. See Column 1 of Figure 5.
Heinrich Niederhausen, A Note on the Enumeration of Diffusion Walks in the First Octant by Their Number of Contacts with the Diagonal, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.3.

See A138350 for a signed version.
Bisections are A000891 and A000888/2.
Cf. A005559, A005560, A005561, A005562, A093768.
Cf. A000108, A005817. Column y=0 of A052174.

[Binomial(n+1, Ceiling(n/2))*Binomial(n, Floor(n/2)) - Binomial(n+1, Ceiling((n-1)/2))*Binomial(n, Floor((n-1)/2)): n in [0..30]]; // Vincenzo Librandi, Sep 30 2015

A:= proc(n,x,y) option remember;
    local j, xpyp, xp,yp, res;
    xpyp:= [[x-1,y],[x+1,y],[x,y-1],[x,y+1]];
    res:= 0;
    for j from 1 to 4 do
      xp:= xpyp[j,1];
      yp:= xpyp[j,2];
      if xp < 0 or xp > yp or xp + yp > n then next fi;
      res:= res + procname(n-1,xp,yp)
    od;
return res
end proc:
A(0,0,0) := 1:
seq(add(add(A(n,x,y), y = x .. n - x), x = 0 .. floor(n/2)), n = 0 .. 50); # Robert Israel, Oct 07 2015

a[n_] := 1/2*Binomial[2*Floor[n/2]+1, Floor[n/2]+1]*CatalanNumber[1/2*(n+Mod[n, 2])]*(Mod[n, 2]+2); Table[a[n]//Abs, {n, 0, 27}] (* Jean-François Alcover, Mar 13 2014 *)

a(n)=binomial(n+1,ceil(n/2))*binomial(n,floor(n/2)) - binomial(n+1,ceil((n-1)/2))*binomial(n,floor((n-1)/2))

from sympy import ceiling as c, binomial
def a(n):
    return binomial(n + 1, c(n/2))*binomial(n, n//2) - binomial(n + 1, c((n - 1)/2))*binomial(n, (n - 1)//2)
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 02 2017

a(n) = C(n+1, ceiling(n/2))*C(n, floor(n/2)) - C(n+1, ceiling((n-1)/2))*C(n, floor((n-1)/2)). - Paul D. Hanna, Apr 16 2004
G.f.: (1/(4x^2))*((16*x^2-1)*(hypergeom([1/2, 1/2],[1],16*x^2)+2*x*(4*x-1)*hypergeom([3/2, 3/2],[2],16*x^2))-2*x+1). - Mark van Hoeij, Oct 13 2009
E.g.f (conjectured): BesselI(1,2*x)*(BesselI(0,2*x)+BesselI(1,2*x))/x. - Benjamin Phillabaum, Feb 25 2011
Conjecture: (2*n+1)*(n+3)*(n+2)*a(n) - 4*(2*n^2+4*n+3)*a(n-1) - 16*n*(2*n+3)*(n-1)*a(n-2) = 0. - R. J. Mathar, Apr 02 2017
Conjecture: (n+3)*(n+2)*a(n) - 4*(n^2+3*n+1)*a(n-1) + 16*(-n^2+n+1)*a(n-2) + 64*(n-1)*(n-2)*a(n-3) = 0. - R. J. Mathar, Apr 02 2017
a(n) = Sum_{k=0..floor(n/2)} n!/(k!*k!*(floor(n/2)-k)!*(floor((n+1)/2)-k)!*(k+1)) (conjectured). - Roger Ford, Aug 04 2017
a(n) = A000108(floor((n+1)/2))*A000108(floor(n/2))*(2*(floor(n/2))+1). - Roger Ford, Nov 15 2019
a(n) = Product_{k=3..n} (4*floor((k-1)/2) + 2) / (floor((k+2)/2)). - Roger Ford, Apr 29 2024

A186420 a(n) = binomial(2n,n)^4.

Original entry on oeis.org

1, 16, 1296, 160000, 24010000, 4032758016, 728933458176, 138735983333376, 27435582641610000, 5588044012339360000, 1165183173971324375296, 247639903129149250277376, 53472066459540320483696896, 11701285507234585729600000000, 2589980371199606611713600000000

Offset: 0

Emanuele Munarini, Feb 21 2011

			G.f.: 4F3({1/2,1/2,1/2,1/2},{1,1,1},256x) where 4F3 is a hypergeometric series.

Seiichi Manyama, Table of n, a(n) for n = 0..417

Cf. binomial(2n,n)^k: A000984 (k=1), A002894 (k=2), A002897 (k=3), this sequence (k=4).

Cf. A000108, A000888, A186414, A186415, A186416, A186418, A186419, A000897, A006480, A008977, A188662.

Table[Binomial[2n,n]^4,{n,0,20}]
Table[Coefficient[Series[HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {1, 1, 1}, 256 x], {x, 0, n}], x, n], {n, 0, 14}] (* Michael De Vlieger, Jul 13 2016 *)

Maxima
```
makelist(binomial(2*n,n)^4,n,0,40);
```

a(n) = A000984(n)^4 = A002894(n)^2.
a(n) = binomial(2*n,n)^4 = ( [x^n](1 + x)^(2*n) )^4 = [x^n](F(x)^(16*n)), where F(x) = 1 + x + 25*x^2 + 1798*x^3 + 183442*x^4 + 22623769*x^5 + 3142959012*x^6 + ... appears to have integer coefficients. For similar results see A000897, A002894, A002897, A006480, A008977 and A188662. - Peter Bala, Jul 14 2016
a(n) ~ 256^n/(Pi*n)^2. - Ilya Gutkovskiy, Jul 13 2016

A186415 a(n) = binomial(2n,n)^3/(n+1).

Original entry on oeis.org

1, 4, 72, 2000, 68600, 2667168, 112698432, 5053029696, 236860767000, 11493303192800, 573327757086656, 29253930349198464, 1521079361361956032, 80361335659444000000, 4304087536829486400000, 233271979857187430688000, 12774642558686527109607000, 706008965215713532853436000, 39337406606398593529683000000

Offset: 0

Emanuele Munarini, Feb 21 2011

Seiichi Manyama, Table of n, a(n) for n = 0..557

Cf. A000108, A002894, A000888, A002897, A186414.

A186415 := proc(n) binomial(2*n,n)^3/(n+1) ; end proc: # R. J. Mathar, Feb 23 2011

Mathematica
```
Table[Binomial[2n,n]^3/(n+1),{n,0,40}]
```

makelist(binomial(2*n,n)^3/(n+1),n,0,40);

G.f.: 3F2(1/2,1/2,1/2;1,2;64x), where 3F2(.,.,.;.,.;.) is a generalized hypergeometric series.
a(n) = A000888(n)*A000984(n). - R. J. Mathar, Feb 23 2011
a(n) ~ 64^n/(Pi^(3/2)*n^(5/2)). - Ilya Gutkovskiy, Nov 01 2016

A186414 a(n) = binomial(2n,n)^3/(n+1)^2.

Original entry on oeis.org

1, 2, 24, 500, 13720, 444528, 16099776, 631628712, 26317863000, 1149330319280, 52120705189696, 2437827529099872, 117006104720150464, 5740095404246000000, 286939169121965760000, 14579498741074214418000

Offset: 0

Emanuele Munarini, Feb 21 2011

Seiichi Manyama, Table of n, a(n) for n = 0..559

Cf. A000108, A002894, A000888, A002897, A186415.

[Binomial(2*n,n)^3/(n+1)^2: n in [0..50]]; // Vincenzo Librandi, Mar 27 2011

Table[Binomial[2n, n]^3/(n + 1)^2, {n, 0, 20}]

makelist(binomial(2*n,n)^3/(n+1)^2,n,0,40);

G.f.: 3F2({1/2, 1/2, 1/2}, {2, 2}, 64x), where 3F2 is a hypergeometric function.

A125558 Central column of triangle A090181.

Original entry on oeis.org

1, 1, 6, 50, 490, 5292, 60984, 736164, 9202050, 118195220, 1551580888, 20734762776, 281248448936, 3863302870000, 53644719852000, 751920156592200, 10626401036545650, 151269944167296900, 2167317913508055000

Offset: 0

Philippe Deléham, Jan 01 2007, Oct 11 2007

[1,6,50,490,5292,...] is a column in triangle of Narayana numbers A001263.
Number of Dyck 2n-paths with exactly n peaks. - Peter Luschny, May 10 2014
For n > 0, number of pairs of non-intersecting lattice paths with steps (1,0), (0,1), where one path goes from (0,0) to (n,n) and the other from (1,0) to (n+1,n). The proof is by switching intersecting path pairs after their first intersection, giving a(n) = binomial(2*n,n)^2 - binomial(2*n+1,n) * binomial(2*n-1,n). - Jeremy Tan, Apr 12 2021

Harvey P. Dale, Table of n, a(n) for n = 0..800

Equals A000888(n)/2 for n>0.

Cf. A090181.

seq(ceil(1/2*(n+1)*((binomial(2*n,n)/(1+n))^2)), n=0..18); # Zerinvary Lajos, Jun 18 2007

CoefficientList[
Series[1 + (HypergeometricPFQ[{1/2, 1/2}, {2}, 16 x] - 1)/(2), {x, 0,
    20}], x]
Join[{1},Table[CatalanNumber[n]^2 (n+1)/2,{n,20}]] (* Harvey P. Dale, Oct 19 2011 *)

a(0)=1, a(n) = Catalan(n)^2*(n+1)/2 = A000108(n)^2*(n+1)/2 for n>0.
a(n) = A090181(2*n, n).
G.f.: 1 + x*3F2( 1, 3/2, 3/2; 2, 3;16 x) = 1 + ( 2F1( 1/2, 1/2; 2;16*x) - 1)/2. - Olivier Gérard, Feb 16 2011
D-finite with recurrence n*(n+1)*a(n) -4*(2*n-1)^2*a(n-1)=0. - R. J. Mathar, Feb 08 2021
a(n) = binomial(2*n,n)^2 - binomial(2*n+1,n) * binomial(2*n-1,n). - Jeremy Tan, Apr 12 2021

A186416 a(n) = binomial(2n,n)^4/(n+1)^3.

Original entry on oeis.org

1, 2, 48, 2500, 192080, 18670176, 2125170432, 270968717448, 37634544090000, 5588044012339360, 875419364366134016, 143310129125665075392, 24338673855047938317568, 4264316875814353400000000, 767401591466550107174400000, 141345980472409642279275210000, 26569505644587874058090478570000

Offset: 0

Emanuele Munarini, Feb 21 2011

Cf. A000108, A002894, A000888, A002897, A186414, A186415.

A186416 := proc(n) binomial(2*n,n)^4/(n+1)^3 ; end proc: # R. J. Mathar, Feb 23 2011

Table[Binomial[2n,n]^4/(n+1)^3,{n,0,40}]

makelist(binomial(2*n,n)^4/(n+1)^3,n,0,40);

G.f.: 4F3(1/2,1/2,1/2,1/2;2,2,2;256*x), where nFm(...;..;.) denotes a generalized hypergeometric series.

a(n) = (A000108(n))^3*A000984(n). - R. J. Mathar, Feb 23 2011

A186229 Expansion of (2F1( (-(1/2), 1/6); (-2/3))( 16 x) -1)/(2*x).

Original entry on oeis.org

1, 14, 182, 2470, 34580, 494760, 7191690, 105793545, 1570873850, 23500272796, 353724885332, 5351515200668, 81313973049064, 1240116577389200, 18973783634054760, 291115203548084370, 4477664537437798980, 69023046543088792440, 1066084706728274263800, 16495237916832025427160, 255635559046076610807120

Offset: 0

Olivier Gérard, Feb 15 2011

Combinatorial interpretation welcome.

Probably a class of paths (Cf. A135404, A000888)

Vincenzo Librandi, Table of n, a(n) for n = 0..200

CoefficientList[Series[(HypergeometricPFQ[{-(1/2), 1/6}, {-(2/3)}, 16 x] - 1)/(2 x), {x, 0, 20}], x]
FullSimplify[Table[-((2^(1/3 + 4 n) (-(4/3))! (-(1/2) + n)! (1/6 + n)!)/(Pi (-(2/3) + n)! (1 + n)!)), {n, 0, 20}]] (* Benedict W. J. Irwin, Jul 12 2016 *)

D-finite with recurrence (n+1)*(3n-2)*a(n) = 4*(6n+1)*(2n-1)*a(n-1). - R. J. Mathar, Jul 11 2012
a(n) ~ 3*GAMMA(2/3)*2^(1/3) * 16^n/(Pi*n^(2/3)). - Vaclav Kotesovec, Aug 13 2013
a(n) = -2^(1/3+4*n)*(-4/3)!*(-1/2+n)!*(1/6+n)!/(Pi*(-2/3+n)!*(1+n)!). - Benedict W. J. Irwin, Jul 12 2016

A186231 Expansion of ( 2F1([-1/4, 1/4]; [-1/2], 16x) - 1 ) / (2x).

Original entry on oeis.org

1, 15, 210, 3003, 43758, 646646, 9657700, 145422675, 2203961430, 33578000610, 513791607420, 7890371113950, 121548660036300, 1877405874732108, 29065024282889672, 450883717216034179, 7007092303604022630, 109069992321755544170, 1700179760011004467468, 26536589497469056215210, 414670662257153823494820

Offset: 0

Olivier Gérard, Feb 15 2011

nonn

Combinatorial interpretation welcome.
Probably a class of paths (Cf. A135404, A000888).
Number of North-East lattice paths from (0,0) to (n,n+1). - Michael D. Weiner, Apr 14 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A186229.

CoefficientList[Series[(HypergeometricPFQ[{-(1/4), 1/4}, {-(1/2)}, 16 x] - 1)/(2 x), {x, 0, 20}], x]

a(n) = A001791(2n+1). - R. J. Mathar, Jul 10 2012

D-finite with recurrence -(n+1)*(2*n-1)*a(n) +2*(4*n-1)*(4*n+1)*a(n-1)=0. - R. J. Mathar, Apr 26 2017

cp's OEIS Frontend

A151392 Duplicate of A000888.

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A013709 a(n) = 4^(2*n+1).

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A005558 a(n) is the number of n-step walks on square lattice such that 0 <= y <= x at each step.

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A186420 a(n) = binomial(2n,n)^4.

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A186415 a(n) = binomial(2n,n)^3/(n+1).

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A186414 a(n) = binomial(2n,n)^3/(n+1)^2.

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A125558 Central column of triangle A090181.

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A186416 a(n) = binomial(2n,n)^4/(n+1)^3.

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A186231 Expansion of ( 2F1([-1/4, 1/4]; [-1/2], 16x) - 1 ) / (2x).