A089835 -id:A089835 - cp's OEIS frontend

A089836 INVERT transform of A089835.

2, 28, 704, 26800, 1404416, 94890112, 7887853568, 779773444864, 89407927009280, 11666949886007296, 1707352344419336192, 276938622991133237248, 49316570352062326636544, 9565558797749164794511360, 2007400581177856844629016576, 453192947995950052758954115072

Offset: 1

Antti Karttunen, Dec 05 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..150

INVERT(A089835); INVERT([seq(A000108(n)*A000108(n)*(n+1)!,n=1..9)]);
a:= proc(n) option remember; local i; `if`(n<1, 1, add(a(n-i) *(2*i)!^2 /i!^3 /(i+1), i=1..n)) end: seq(a(n), n=1..20); # Alois P. Heinz, Apr 01 2009

a[n_] := a[n] = If[n < 1, 1, Sum[a[n - i] *(2 i)!^2 /i!^3 /(i + 1), {i, 1, n}]]; Array[a, 20] (* Jean-François Alcover, Mar 03 2016, adapted from Maple *)

a(n) ~ 16^n * n! / (Pi*n^2). - Vaclav Kotesovec, Sep 05 2014

More terms from Alois P. Heinz, Apr 01 2009

A000888 a(n) = (2n)!^2 / ((n+1)!n!^3).

Original entry on oeis.org

1, 2, 12, 100, 980, 10584, 121968, 1472328, 18404100, 236390440, 3103161776, 41469525552, 562496897872, 7726605740000, 107289439704000, 1503840313184400, 21252802073091300, 302539888334593800, 4334635827016110000, 62464383654579522000, 904841214653480504400

Offset: 0

N. J. A. Sloane

a(n) is the number of walks of 2n unit steps North, East, South, or West, starting at the origin, bounded above by y=x, below by y=-x and terminating on the ray y = x >= 0. Example: a(1) counts EN, EW; a(2) counts ESNN, ESNW, ENSN, ENSW, ENEN, ENEW, EENN, EENW, EEWN, EEWW, EWEN, EWEW. - David Callan, Oct 11 2005
Bijective proof: given such an NESW walk, construct a pair (P_1, P_2) of lattice paths of upsteps U=(1,1) and downsteps D=(1,-1) as follows. To get P_1, replace each E and S with U and each W and N with D. To get P_2, replace each N and E with U and each S and W with D. For example, EENSNW -> (UUDUDD, UUUDUD). This mapping is 1-to-1 and its range is the Cartesian product of the set of Dyck n-paths and the set of nonnegative paths of length 2n. The Dyck paths are counted by the Catalan number C_n (A000108) and the nonnegative paths are counted (see for example the Callan link) by the central binomial coefficient binomial(2n,n) (A000984). So this is a bijection from these NESW walks to a set of size C_n*binomial(2n,n) = a(n). - David Callan, Sep 18 2007
If A is a random matrix in USp(4) (4 X 4 complex matrices that are unitary and symplectic), then a(n) = E[(tr(A^3))^{2n}]. - Andrew V. Sutherland, Apr 01 2008
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,-1), (-1,1), (1,-1), (1,1)}. - Manuel Kauers, Nov 18 2008
a(n) is equal to the n-th moment of the following positive function defined on x in (0,16), in Maple notation: (EllipticK(sqrt(1-x/16)) - EllipticE(sqrt(1-x/16)))/(Pi^2*sqrt(x)). This is the solution of the Hausdorff moment problem and thus it is unique. - Karol A. Penson, Feb 11 2011
The partial sums of a(n)/A013709(n) absolutely converge to 1/Pi. - Ralf Steiner, Jan 21 2016

			G.f.: 1 + 2*x + 12*x^2 + 100*x^3 + 980*x^4 + 10584*x^5 + 121968*x^6 + ...

E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 93.
T. M. MacRobert, Functions of a Complex Variable, 4th ed., Macmillan & Co., London, 1958, p. 177.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Marco S. Bianchi, Protected and uniformly transcendental, arXiv:2306.06239 [hep-th], 2023.
M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
David Callan, Bijections for the identity 4^n = ... .
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.
Ralf Steiner, Beispiele zur modifizierten Wallis-Lambert-Reihe, 2016.

Cf. A000108, A002894, A088538, A089835, A125558.

[(Factorial(2*n))^2/(Factorial(n))^4/(n+1): n in [0..20]]; // Vincenzo Librandi, Aug 15 2011

[seq(binomial(2*n,n)^2/(n+1),n=0..17)]; # Zerinvary Lajos, May 27 2006

f[n_] := Binomial[2 n, n]^2/(n + 1); Array[f, 18, 0]  (* Robert G. Wilson v *)
a[ n_] := SeriesCoefficient[ (1/8) (EllipticE[ 16 x] - (1 - 16 x) EllipticK[ 16 x]) / (Pi/2), {x, 0, n + 1}]; (* Michael Somos, Jan 23 2012 *)

{a(n) = if( n<0, 0, (2*n)!^2 / n!^4 / (n+1))}; /* Michael Somos, Sep 11 2005 */

G.f.: 1/4*((16*x-1)*EllipticK(4*x^(1/2)) + EllipticE(4*x^(1/2)))/x/Pi. - Vladeta Jovovic, Oct 12 2003
Given G.f. A(x), y = x*A(x) satisfies y = y'' * (1 - 16*x) * x/4. - Michael Somos, Sep 11 2005
a(n) = binomial(2*n,n)^2/(n+1). - Zerinvary Lajos, May 27 2006
G.f.: 2F1(1/2,1/2;2;16*x). - Paul Barry, Sep 03 2008
a(n) = 2*A125558(n) (n >= 1). - Olivier Gérard, Feb 16 2011
A002894(n) = (n+1) * a(n). A001246(n) = a(n) / (n+1). A089835(n) = n! * a(n). - Michael Somos, May 12 2012
G.f.: 1 + 4*x/(G(0)-4*x) where G(k) = 4*x*(2*k+1)^2 + (k+1)*(k+2) - 4*x*(k+1)*(k+2)*(2*k+3)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 30 2012
D-finite with recurrence: (n+1)*(n+2)*a(n+1) = 4*(2*n+1)^2*a(n). - Vaclav Kotesovec, Sep 11 2012
a(n) = C(n)*binomial(2*n,n) = Sum_{k=0..2*n} binomial(2*n,k)*C(k)*C(2*n-k) where C(k) are Catalan numbers (A000108), see Prodinger. - Michel Marcus, Nov 19 2019
Sum_{n>=0} a(n)/16^n = 4/Pi (A088538). - Amiram Eldar, May 06 2023

A089833 a(n) = A000108(n)*(A000142(n+1)-1).

Original entry on oeis.org

0, 1, 10, 115, 1666, 30198, 665148, 17296851, 518916970, 17643220738, 670442556004, 28158587998814, 1295295050441588, 64764752531737100, 3497296636751245560, 202843204931717665155, 12576278705767060962330

Offset: 0

Antti Karttunen, Dec 05 2003

nonn

The first column of A089831.

Cf. A000108, A001813, A089835.

a(n) = A001813(n) - A000108(n).

A350266 Triangle read by rows. T(n, k) = binomial(n, k) * n! / (n - k + 1)! if k >= 1, if k = 0 then T(n, k) = k^n. T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 9, 6, 0, 4, 24, 48, 24, 0, 5, 50, 200, 300, 120, 0, 6, 90, 600, 1800, 2160, 720, 0, 7, 147, 1470, 7350, 17640, 17640, 5040, 0, 8, 224, 3136, 23520, 94080, 188160, 161280, 40320, 0, 9, 324, 6048, 63504, 381024, 1270080, 2177280, 1632960, 362880

Offset: 0

Peter Luschny, Jan 09 2022

			Table starts:
[0] 1;
[1] 0, 1;
[2] 0, 2,   2;
[3] 0, 3,   9,    6;
[4] 0, 4,  24,   48,    24;
[5] 0, 5,  50,  200,   300,    120;
[6] 0, 6,  90,  600,  1800,   2160,     720;
[7] 0, 7, 147, 1470,  7350,  17640,   17640,    5040;
[8] 0, 8, 224, 3136, 23520,  94080,  188160,  161280,   40320;
[9] 0, 9, 324, 6048, 63504, 381024, 1270080, 2177280, 1632960, 362880;

A350267 (row sums), A000142 (main diagonal), A074143 (subdiagonal), A006002 (column 2), A089835 (central terms).

T := (n, k) -> ifelse(k = 0, k^n, binomial(n, k)^2 * k! / (n - k + 1)):
seq(seq(T(n, k), k = 0..n), n = 0..9);

T[n_, 0] := Boole[n == 0]; T[n_, k_] := Binomial[n, k]^2 * k!/(n - k + 1); Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jan 09 2022 *)

T(n, k) = binomial(n, k)^2 * k! / (n - k + 1) if k >= 1.

cp's OEIS Frontend

A089836 INVERT transform of A089835.

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A000888 a(n) = (2*n)!^2 / ((n+1)!*n!^3).

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A089833 a(n) = A000108(n)*(A000142(n+1)-1).

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Formula

A350266 Triangle read by rows. T(n, k) = binomial(n, k) * n! / (n - k + 1)! if k >= 1, if k = 0 then T(n, k) = k^n. T(n, k) for 0 <= k <= n.

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Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A000888 a(n) = (2n)!^2 / ((n+1)!n!^3).