A002736 -id:A002736 - cp's OEIS frontend

A110609 a(n) = n * binomial(2*n, n-1).

0, 1, 8, 45, 224, 1050, 4752, 21021, 91520, 393822, 1679600, 7113106, 29953728, 125550100, 524190240, 2181340125, 9051563520, 37467344310, 154754938800, 637982011590, 2625648168000, 10789623755820, 44277560801760, 181478535620850, 742984788858624, 3038716500907500

Offset: 0

Paul Barry, Jul 30 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.

Cf. A000108, A002390, A002736, A253487.

Column k=1 of A110608.

[0] cat [((4*n+4)*(2*n+1)*Binomial(2*n, n)/(n+2))/2: n in [0..25]]; // Vincenzo Librandi, Jan 09 2015

with(combinat):with(combstruct):a[0]:=0:for n from 1 to 30 do a[n]:=sum((count(Composition(n*2+1),size=n)),j=1..n) od: seq(a[n], n=0..22); # Zerinvary Lajos, May 09 2007
a:=n->sum(sum(binomial(2*n,n)/(n+1), j=1..n),k=1..n): seq(a(n), n=0..22); # Zerinvary Lajos, May 09 2007
series(simplify(x*diff(x*diff((1-sqrt(1-4*x))/(2*x), x), x)), x, 20):
seq(coeff(%, x, k), k=0..18); # Karol A. Penson, Apr 25 2025

Table[CatalanNumber[n]*n^2, {n, 0, 22}] (* Zerinvary Lajos, Jul 08 2009 *)
CoefficientList[Series[x (1 / x^2 - (1 - 6 x + 4 x^2) / ((1 - 4 x)^(3/2) x^2)) / 2, {x, 0, 30}], x] (* Vincenzo Librandi, Jan 09 2015 *)

for(n=0,25, print1(n*binomial(2*n,n-1), ", ")) \\ G. C. Greubel, Sep 01 2017

a(n) = n^2*binomial(2*n, n)/(n+1) = n^2*A000108(n) = A002736(n)/(n+1).
G.f.: -(2*x*(2*x+2*sqrt(1-4*x)-3) - sqrt(1-4*x) + 1)/(2*sqrt((1 - 4*x)^3)*x). - Marco A. Cisneros Guevara, Jul 23 2011; amended by Georg Fischer, Apr 09 2020
(n+1)*(10*n-7)*a(n)+2*n*(5*n-88)*a(n-1) -4*(25*n-22)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 07 2012
From Ilya Gutkovskiy, Jan 20 2017: (Start)
E.g.f.: x*(BesselI(0,2*x) + 2*BesselI(1,2*x) + BesselI(2,2*x))*exp(2*x).
a(n) ~ 4^n*sqrt(n)/sqrt(Pi).
Sum_{n>=1} 1/a(n) = Pi*(2*sqrt(3) + Pi)/18 = 1.152911143694148... (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (2/sqrt(5))*log(phi) + 2*log(phi)^2, where log(phi) = A002390. - Amiram Eldar, Feb 20 2021
G.f.: (x*(d/dx))^2 [g.f. of A000108]. - Karol A. Penson, Apr 25 2025

A189765 Triangle in which row n has the n(n+1)/2 elements of the lower triangular part of the inverse of the n-th order Hilbert matrix.

Original entry on oeis.org

1, 4, -6, 12, 9, -36, 192, 30, -180, 180, 16, -120, 1200, 240, -2700, 6480, -140, 1680, -4200, 2800, 25, -300, 4800, 1050, -18900, 79380, -1400, 26880, -117600, 179200, 630, -12600, 56700, -88200, 44100, 36, -630, 14700, 3360, -88200, 564480, -7560, 211680

Offset: 1

T. D. Noe, May 02 2011

The n-th order Hilbert matrix has elements h(i,j) = 1/(i+j-1) for 1 <= i,j <=n. Only the lower triangular matrix is shown because the Hilbert matrix and its inverse are symmetric. The n-th row begins with n^2 and ends with A000515(n+1).
The sums of select rows of the inverse matrix are sequences A002457, A002736, A002738, A007531, and A054559.
The largest magnitude in the matrix is A210356(n). - T. D. Noe, Mar 28 2012
The sum of the elements of the n-th matrix is n^2. - T. D. Noe, Apr 02 2012

			Row 3 is 9, -36, 192, 30, -180, 180 which corresponds to the inverse
  9  -36   30
-36  192 -180
30 -180  180

T. D. Noe, Rows n = 1..25, flattened
Eric W. Weisstein, MathWorld: Hilbert Matrix

Cf. A002457, A002736, A002738, A005249 (determinant), A007531, A054559, A189766 (trace).

lowerTri[m_List] := Module[{n = Length[m]}, Flatten[Table[Take[m[[i]], i], {i, n}]]]; Flatten[Table[lowerTri[Inverse[HilbertMatrix[n]]], {n, 6}]]

a(n,i,j) = (-1)^(i+j) (i+j-1) binomial(n+i-1, n-j) binomial(n+j-1, n-i) binomial(i+j-2, i-1)^2 is the (i,j) element of the inverse of the n-th Hilbert matrix.

A069121 a(n) = n^4*binomial(2n,n).

Original entry on oeis.org

0, 2, 96, 1620, 17920, 157500, 1197504, 8240232, 52715520, 318995820, 1847560000, 10328229912, 56073378816, 297051536600, 1541119305600, 7852824450000, 39392404439040, 194905125100620, 952671403252800

Offset: 0

Benoit Cloitre, Apr 07 2002

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 386.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A002736.

with(combinat):for n from 0 to 18 do printf(`%d, `,n^3*sum(binomial(2*n, n), k=1..n)) od: # Zerinvary Lajos, Mar 13 2007

Table[n^4*Binomial[2 n, n], {n, 0, 18}] (* or *)
CoefficientList[Series[2 x (1 + 30 x + 72 x^2 + 8 x^3)/(1 - 4 x)^(9/2), {x, 0, 18}], x] (* Michael De Vlieger, Feb 07 2017 *)

PARI
```
a(n)=if(n<1,0,n^4*binomial(2*n,n))
```

Sum_{n>=1} 1/a(n) = 17*Pi^4/3240. (Comtet, 1974)
a(n) = a(n-1)*(4*n-2)*n^3/(n-1)^4, n>1. - Michael Somos, Apr 18 2003
Equals A002736*n^2. - Zerinvary Lajos, May 28 2006
From Ilya Gutkovskiy, Feb 07 2017: (Start)
G.f.: 2*x*(1 + 30*x + 72*x^2 + 8*x^3)/(1 - 4*x)^(9/2).
a(n) ~ 4^n*n^(7/2)/sqrt(Pi). (End)

A119553 Binomial(binomial(2n,n)n^2,n).

Original entry on oeis.org

1, 2, 276, 955860, 65212649320, 82571843488838760, 1880695497510691320340728, 754603377505528950689544452061864, 5254517954094415196615118245270696186523600

Offset: 0

Zerinvary Lajos, May 30 2006

Cf. A002736.

[seq (binomial(binomial(2*n,n)*n^2,n),n=0..10)];

A131972 Sum of all n-digit Apery numbers.

Original entry on oeis.org

2, 24, 180, 7420, 33264, 991848, 3938220, 103832872, 389398464, 9620555000, 34901442000, 828288777420, 2940343837200, 67898251759800, 237371722628040, 5373868753340880, 97581248745060600, 335240928272918304, 7415892272293658608, 25286571126114014640, 553714770886681187168

Offset: 1

Parthasarathy Nambi, Oct 06 2007

			Sum of all 1-digit Apery numbers is 0 + 2 = 2.
Sum of all 2-digit Apery numbers is 24
Sum of all 3-digit Apery numbers is 180.

Cf. A002736.

digNum[n_] := Length @ IntegerDigits[n]; apery[n_] := n^2 * Binomial[2n, n]; digCount = 0; sum = 0; cumsum = {}; Do[a = apery[n]; If[digNum[a] > digCount, digCount++; AppendTo[cumsum, sum]]; sum += a, {n, 0, 35}]; Differences[cumsum] (* Amiram Eldar, Nov 30 2019 *)

More terms from Amiram Eldar, Nov 30 2019

A339470 Decimal expansion of log(phi)^2, where phi is the golden ratio (A002390^2).

Original entry on oeis.org

2, 3, 1, 5, 6, 4, 8, 2, 0, 5, 7, 7, 1, 9, 4, 3, 9, 2, 4, 9, 6, 9, 2, 9, 0, 7, 1, 2, 3, 1, 5, 3, 2, 7, 6, 0, 0, 1, 6, 4, 0, 6, 3, 5, 0, 0, 4, 9, 2, 9, 8, 8, 7, 0, 8, 1, 5, 3, 0, 1, 2, 2, 8, 6, 8, 9, 7, 9, 5, 3, 4, 5, 5, 6, 6, 9, 6, 1, 8, 1, 2, 9, 8, 5, 0, 5, 4

Offset: 0

Robert Bilinski, Dec 06 2020

			0.2315648205771943924969290712315327600164063500492988708153012286...

Travis Sherman, Summation of Glaisher- and Apéry-like Series, University of Arizona, 2000.
Index entries for transcendental numbers

Cf. A002390, A001622, A086467.

RealDigits[Log[GoldenRatio]^2, 10, 100][[1]] (* Amiram Eldar, Dec 06 2020 *)

asinh(1/2)^2 \\ Michel Marcus, Dec 06 2020

Equals arcsinh(1/2)^2 = A002390^2.
Equals (1/2)*Sum_{k>=1} ((k!)^2*(-1)^(k+1))/((2*k)!*k^2) = A086467/2.
Equals (1/3)*(zeta(2) - Sum_{k>=1} ((k!)^2*(-1)^k)/((2*k)!*(2*k+1)^2)).
Equals (1/2)*Sum_{k>=1} (-1)^(k+1)/A002736(k).

cp's OEIS Frontend

A110609 a(n) = n * binomial(2*n, n-1).

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A189765 Triangle in which row n has the n(n+1)/2 elements of the lower triangular part of the inverse of the n-th order Hilbert matrix.

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A069121 a(n) = n^4*binomial(2n,n).

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A119553 Binomial(binomial(2*n,n)*n^2,n).

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A131972 Sum of all n-digit Apery numbers.

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A339470 Decimal expansion of log(phi)^2, where phi is the golden ratio (A002390^2).

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Mathematica

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A119553 Binomial(binomial(2n,n)n^2,n).