A089460 Triangle, read by rows, of coefficients for the second iteration of the hyperbinomial transform.

1, 2, 1, 8, 4, 1, 50, 24, 6, 1, 432, 200, 48, 8, 1, 4802, 2160, 500, 80, 10, 1, 65536, 28812, 6480, 1000, 120, 12, 1, 1062882, 458752, 100842, 15120, 1750, 168, 14, 1, 20000000, 8503056, 1835008, 268912, 30240, 2800, 224, 16, 1, 428717762, 180000000, 38263752, 5505024, 605052, 54432, 4200, 288, 18, 1

Offset: 0

Author

Paul D. Hanna, Nov 05 2003

Keywords

• nonn
• tabl

Comments

Equals the matrix square of A088956 when treated as a lower triangular matrix. The 2nd hyperbinomial transform of a sequence {b} is defined to be the sequence {d} given by d(n) = Sum_{k=0..n} T(n,k)*b(k), where T(n,k) = 2*(n-k+2)^(n-k-1)*C(n,k). Given a table in which the n-th row is the n-th binomial transform of the first row, then the 2nd hyperbinomial transform of any diagonal results in the diagonal located 2 diagonals lower in the table.

Examples

			Rows begin:
  {1},
  {2,1},
  {8,4,1},
  {50,24,6,1},
  {432,200,48,8,1},
  {4802,2160,500,80,10,1},
  {65536,28812,6480,1000,120,12,1},
  {1062882,458752,100842,15120,1750,168,14,1},..
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A089461(row sums), A089462(diagonal), A089463, A088956.

Programs

Mathematica

Join[{1}, Table[Binomial[n, k]*2*(n - k + 2)^(n - k - 1), {n, 1, 49}, {k, 0, n}]] // Flatten (* G. C. Greubel, Nov 18 2017 *)

PARI

for(n=0,10, for(k=0,n, print1(2*(n-k+2)^(n-k-1)*binomial(n,k), ", "))) \\ G. C. Greubel, Nov 18 2017

Formula

T(n, k) = 2*(n-k+2)^(n-k-1)*C(n, k).

E.g.f.: exp(x*y)*(-LambertW(-y)/y)^2.

Note: (-LambertW(-y)/y)^2 = Sum_{n>=0} 2*(n+2)^(n-1)*y^n/n!.