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

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

Author

Peter Luschny, Oct 17 2014

Keywords

• nonn
• tabl

Comments

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)

Examples

			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;
		

Links

• 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.

Crossrefs

Cf. A247499 (row sums), A008297.

Cf. A000142, A000217, A001710.

Cf. A204515 (central terms), A105278, A004736.

Cf. A090181, A001263.

Programs

Haskell

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

Magma

/* 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

Maple

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):

Mathematica

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 *)

Formula

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)

Extensions

Name updated by Peter Luschny, Jan 09 2022