A204515 -id:A204515 - cp's OEIS frontend

A248045 (2(n-1))! (2n-1)! / (n (n-1)!^3).

1, 6, 120, 4200, 211680, 13970880, 1141620480, 111307996800, 12614906304000, 1629845894476800, 236475822507724800, 38072607423743692800, 6735922851893114880000, 1299070835722243584000000, 271245990498804460339200000, 60962536364606302461235200000

Offset: 1

Reinhard Zumkeller, Sep 30 2014

nonn

Central terms in triangles of Lah numbers: a(n) = - A008297(2*n-1,n) = A105278(2*n-1,n) = A000891(n-1)*A000142(n) = A000894(n-1)*A000142(n-1).

a(n) = n * A204515(n-1). - Reinhard Zumkeller, Oct 19 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Cf. A000142, A000891, A000894, A008297, A105278, A204515.

Cf. A187535 (Central Lah numbers).

Haskell
```
a248045 n = a000891 (n - 1) * a000142 n
```

n*a(n) = 4*(2*n-1)*(2*n-3)*a(n-1). - R. J. Mathar, Oct 07 2014

A247500 Triangle read by rows: T(n, k) = n!*binomial(n + 1, k)/(k + 1)!, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 12, 6, 1, 24, 60, 40, 10, 1, 120, 360, 300, 100, 15, 1, 720, 2520, 2520, 1050, 210, 21, 1, 5040, 20160, 23520, 11760, 2940, 392, 28, 1, 40320, 181440, 241920, 141120, 42336, 7056, 672, 36, 1, 362880, 1814400, 2721600, 1814400, 635040, 127008, 15120, 1080, 45, 1

Offset: 0

Peter Luschny, Oct 17 2014

An alternative definition would have been: (n-k)!*N(n,k) where N(n,k) are the little Narayana numbers A090181(n,k). This adds a first column (1,0,0,...) to the triangle and amounts to (Gamma(n)*Gamma(n+1))/(Gamma(k)*Gamma(k+1)*Gamma(n-k+2)). - Peter Luschny, Jun 18 2015
From Peter Bala, Sep 03 2023: (Start)
Let E(y) = Sum_{n >= 0} y^n/(n+1)!. Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence n!*(n+1)! as defined in Wang and Wang.
Let B(y) = Sum_{n >= 0} y^n/(n!*(n+1)!) = 1/sqrt(y)*BesselI(1,2*sqrt(y)). A generating function for the triangle is E(y)*B(x*y) = 1 + (1 + x)*y/(1!*2!) + (2 + 3*x + x^2)*y^2/(2!*3!) + (6 + 12*x + 6*x^2 + x^3)*y^3/(3!*4!) + .... Cf. A105278 with a generating function exp(y)*B(x*y).
The n-th power of this array has a generating function E(y)^n*B(x*y). In particular, the matrix inverse has a generating function B(x*y)/E(y). (End)

			Triangle begins:
                      1;
                   1,    1;
                2,    3,    1;
             6,   12,    6,    1;
         24,   60,   40,   10,    1;
     120,  360,  300,  100,   15,    1;
  720, 2520, 2520, 1050,  210,   21,    1;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
W. Wang and T. Wang, Generalized Riordan arrays, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Cf. A247499 (row sums), A008297.
Cf. A000142, A000217, A001710.
Cf. A204515 (central terms), A105278, A004736.
Cf. A090181, A001263.

a247500 n k = a247500_tabl !! n !! k
a247500_row n = a247500_tabl !! n
a247500_tabl = zipWith (zipWith div) a105278_tabl a004736_tabl
-- Reinhard Zumkeller, Oct 19 2014

/* triangle */ [[Factorial(n)/Factorial(k) * Binomial(n+2, k+1) /(n+2): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 18 2014

T := (n,k) -> ((k+1)*(n+1)*GAMMA(n+1)^2)/(GAMMA(k+2)^2*GAMMA(n-k+2));
A247500 := (n, k) -> (n!/(k+1)!)*binomial(n + 1, k):

Table[((k + 1) (n + 1) Gamma[n + 1]^2)/(Gamma[k + 2]^2*
Gamma[n - k + 2]), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jun 19 2015 *)

T(n, k) = ((k+1)*(n+1)*Gamma(n+1)^2)/(Gamma(k+2)^2 *Gamma(n-k+2)). (original name)
T(n, k) = (n!/k!)*C(n+2, k+1)/(n+2).
T(n, 0) = A000142(n).
T(n, n-1) = A000217(n).
T(n+1, 1) = A001710(n+2).
Sum_{k=0..n} T(n, k) = A247499(n).
L(n+1, k+1) = T(n-1, k)*P(n) for n>=1 and 0<=k<=n; here L(n,k) denote the unsigned Lah numbers and P(n) the pronic numbers. - Peter Luschny, Oct 18 2014
T(n,k) = A105278(n+1,k+1) / (n+1-k), k=0..n. - Reinhard Zumkeller, Oct 19 2014
From Peter Bala, May 24 2023: (Start)
Triangle equals A164652 * A008277 (assuming the same offset for the three triangles).
This is equivalent to the Stirling number identity Sum_{i = 0..n} (n+1)!/(i+1)!* binomial(n,i)*Stirling1(i+1,k) = (-1)^(n+k+1)*Stirling1(n+1,k) for n, k >= 0. (End)

Name updated by Peter Luschny, Jan 09 2022

cp's OEIS Frontend

A248045 (2(n-1))! (2n-1)! / (n (n-1)!^3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Formula

A247500 Triangle read by rows: T(n, k) = n!*binomial(n + 1, k)/(k + 1)!, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Formula

Extensions

cp's OEIS Frontend

A248045 (2*(n-1))! * (2*n-1)! / (n * (n-1)!^3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Formula

A247500 Triangle read by rows: T(n, k) = n!*binomial(n + 1, k)/(k + 1)!, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Formula

Extensions

A248045 (2(n-1))! (2n-1)! / (n (n-1)!^3).