A052704 -id:A052704 - cp's OEIS frontend

A151403 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of 2*n steps taken from {(-1, 0), (-1, 1), (1, 0), (1, 1)}.

1, 4, 32, 320, 3584, 43008, 540672, 7028736, 93716480, 1274544128, 17611882496, 246566354944, 3489862254592, 49855175065600, 717914520944640, 10409760553697280, 151860036312760320, 2227280532587151360, 32823081532863283200, 485781606686376591360, 7217326727911880785920

Offset: 0

Manuel Kauers, Nov 18 2008

Essentially the same as A052704. - R. J. Mathar, Nov 27 2008
From Joerg Arndt, Oct 22 2012: (Start)
Number of strings of length 2*n of four different types of balanced parentheses.
The number of strings of length 2*n of t different types of balanced parentheses is given by t^n * A000108(n): there are n opening parentheses in the strings, giving t^n choices for the type (the closing parentheses are chosen to match). (End)
Number of Dyck paths of length 2n in which the step U=(1,1) comes in 4 colors. - José Luis Ramírez Ramírez, Jan 31 2013

Richard P. Stanley, Catalan Numbers, Cambridge, 2015, p. 106.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Mireille Bousquet-Mélou and Marni Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
Alon Regev, Enumerating Triangulations by Parallel Diagonals, Journal of Integer Sequences, Vol. 15 (2012), Article 12.8.5; arXiv preprint, arXiv:1208.3915 [math.CO], 2012.

Cf. A000108, A052704, A085880, A269796.

[4^n * Catalan(n): n in [0..20]]; // Vincenzo Librandi, Oct 24 2012

A151403_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 4*(a[w-1]+add(a[j]*a[w-j-1],j=1..w-1)) od; convert(a,list) end: A151403_list(20); # Peter Luschny, May 19 2011
seq(4^n*(2*n)!*coeff(series(hypergeom([],[2],x^2),x,2*n+2),x,2*n),n=0..20); # Peter Luschny, Jan 31 2015

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[-1 + i, j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[Sum[aux[0, k, 2 n], {k, 0, 2 n}], {n, 0, 25}]

A151403 = lambda n: 4^n*hypergeometric([1-n,-n],[2],1)
[Integer(A151403(n).n()) for n in range(21)] # Peter Luschny, Sep 22 2014

a(n) = 4^n*A000108(n). - Philippe Deléham, Feb 01 2009
a(n) = Integral_{x=-2..2} (2*x)^(2*n)*sqrt((2-x)*(2+x))/(2*Pi) dx. - Peter Luschny, Sep 11 2011
E.g.f.: KummerM(1/2, 2, 16*x). - Peter Luschny, Aug 26 2012
G.f.: 2/(1 + sqrt(1-16*x)) = 1/U(0) where U(k) = 1 - 4*x/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 30 2012
G.f.: c(4*x) with c(x) the o.g.f. of A000108 (Catalan). - Philippe Deléham, Nov 15 2013
a(n) = sum{k=0..n} A085880(n,k)*3^k. - Philippe Deléham, Nov 15 2013
a(n) = 4^n*hypergeom([1-n,-n],[2],1). - Peter Luschny, Sep 22 2014
a(n) = 4^n*(2*n)!*[x^(2*n)]hypergeom([],[2],x^2). - Peter Luschny, Jan 31 2015
a(n) ~ 2^(4*n+2)/((2*n+1)*sqrt(Pi*(4*n+5))). - Peter Luschny, Jan 31 2015
D-finite with recurrence: (n+1)*a(n) +8*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Feb 21 2020
Sum_{n>=0} 1/a(n) = 88/75 + 128*arctan(1/sqrt(15)) / (75*sqrt(15)). - Vaclav Kotesovec, Nov 23 2021
Sum_{n>=0} (-1)^n/a(n) = 248/289 - 384*arctanh(1/sqrt(17)) / (289*sqrt(17)). - Amiram Eldar, Jan 25 2022
G.f.: 1/(1-4*x/(1-4*x/(1-4*x/(1-4*x/(1-4*x/(1-4*x/(1-4*x/(1-4*x/(1-...)))))))))(continued fraction), cf. g.f. by Sergei N. Gladkovskii. - Nikolaos Pantelidis, Nov 21 2022
a(n) = 4*A269796(n-1) for n>0. - Hugo Pfoertner, Oct 04 2024

A354735 a(0) = a(1) = 1; a(n) = 4 * Sum_{k=0..n-2} a(k) * a(n-k-2).

Original entry on oeis.org

1, 1, 4, 8, 36, 96, 416, 1312, 5504, 19200, 79168, 293888, 1203712, 4648448, 19027968, 75411456, 309487616, 1248411648, 5144133632, 21011775488, 86971449344, 358540509184, 1490753372160, 6189315784704, 25843660619776, 107902536122368, 452308820819968, 1897178275512320

Offset: 0

Ilya Gutkovskiy, Jun 04 2022

nonn

Cf. A007477, A052704, A151403, A354733, A354734, A354736.

a[0] = a[1] = 1; a[n_] := a[n] = 4 Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 27}]
nmax = 27; CoefficientList[Series[(1 - Sqrt[1 - 16 x^2 (1 + x)])/(8 x^2), {x, 0, nmax}], x]

G.f. A(x) satisfies: A(x) = 1 + x + 4 * (x * A(x))^2.
G.f.: (1 - sqrt(1 - 16 * x^2 * (1 + x))) / (8 * x^2).
a(n) ~ sqrt((2+3*r)*(1+r)) / (sqrt(Pi) * n^(3/2) * r^n), where r = 2*cos(arccos(-5/32)/3)/3 - 1/3. - Vaclav Kotesovec, Jun 04 2022

A178657 Irregular triangle: the coefficient [x^k] of the polynomial (1-x)^(2n-1) Sum_{s>=0} A001263(n+2s,2s+1)*x^s in row n >= 1 and column k >= 0.

Original entry on oeis.org

1, 1, 3, 1, 15, 15, 1, 1, 43, 161, 105, 10, 1, 96, 855, 1680, 855, 96, 1, 1, 185, 3191, 13387, 17655, 7623, 945, 21, 1, 323, 9570, 72254, 188188, 188188, 72254, 9570, 323, 1, 1, 525, 24675, 302359, 1345605, 2499861, 2036125, 715725, 99414, 4410, 36, 1, 808

Offset: 1

Roger L. Bagula, Jun 01 2010

Row sums are 1, 4, 32, 320, 3584, 43008, 540672, 7028736, 93716480, 1274544128, ... (see A151403, A052704).

The sequence is the Narayana number analog of A034839.

			1;
1, 3;
1, 15, 15, 1;
1, 43, 161, 105, 10;
1, 96, 855, 1680, 855, 96, 1;
1, 185, 3191, 13387, 17655, 7623, 945, 21;
1, 323, 9570, 72254, 188188, 188188, 72254, 9570, 323, 1;
1, 525, 24675, 302359, 1345605, 2499861, 2036125, 715725, 99414, 4410, 36;
1, 808, 56896, 1055320, 7329975, 22338816, 32152848, 22338816, 7329975, 1055320, 56896, 808, 1;

Cf. A001263, A034839.

A001263 := proc(n,k) if n <=0 or k <=0 then 0 ; elif k > n then 0 ; else binomial(n-1,k-1)*binomial(n,k-1)/k ; end if; end proc:
A178657 := proc(n,k) (1-x)^(2*n-1)*add(A001263(n+2*l,2*l+1)*x^l,l=0..20) ; expand(%) ; coeftayl(%,x=0,k) ; end proc: # R. J. Mathar, Aug 30 2011

p[x_, n_] = (1 - x)^(2*n - 1)*Sum[(Binomial[2*k + n, 2*k] Binomial[ 2*k + n, 1 + 2*k]/(2*k + n))*x^k, {k, 0, Infinity}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 2, 10}];
Flatten[%]

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A151403 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of 2*n steps taken from {(-1, 0), (-1, 1), (1, 0), (1, 1)}.

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A354735 a(0) = a(1) = 1; a(n) = 4 * Sum_{k=0..n-2} a(k) * a(n-k-2).

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A178657 Irregular triangle: the coefficient [x^k] of the polynomial (1-x)^(2n-1) Sum_{s>=0} A001263(n+2s,2s+1)*x^s in row n >= 1 and column k >= 0.

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cp's OEIS Frontend

A151403 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of 2*n steps taken from {(-1, 0), (-1, 1), (1, 0), (1, 1)}.

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Magma

Maple

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A354735 a(0) = a(1) = 1; a(n) = 4 * Sum_{k=0..n-2} a(k) * a(n-k-2).

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A178657 Irregular triangle: the coefficient [x^k] of the polynomial (1-x)^(2*n-1) * Sum_{s>=0} A001263(n+2*s,2*s+1)*x^s in row n >= 1 and column k >= 0.

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A178657 Irregular triangle: the coefficient [x^k] of the polynomial (1-x)^(2n-1) Sum_{s>=0} A001263(n+2s,2s+1)*x^s in row n >= 1 and column k >= 0.