A089463 Triangle, read by rows, of coefficients for the third iteration of the hyperbinomial transform.

1, 3, 1, 15, 6, 1, 108, 45, 9, 1, 1029, 432, 90, 12, 1, 12288, 5145, 1080, 150, 15, 1, 177147, 73728, 15435, 2160, 225, 18, 1, 3000000, 1240029, 258048, 36015, 3780, 315, 21, 1, 58461513, 24000000, 4960116, 688128, 72030, 6048, 420, 24, 1, 1289945088

Offset: 0

Author

Paul D. Hanna, Nov 05 2003

Keywords

• nonn
• tabl

Comments

Equals the matrix cube of A088956 when treated as a lower triangular matrix. The 3rd 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) = 3*(n-k+3)^(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 3rd hyperbinomial transform of any diagonal results in the 3rd diagonal lower in the table.

Examples

			Rows begin:
  {1},
  {3,1},
  {15,6,1},
  {108,45,9,1},
  {1029,432,90,12,1},
  {12288,5145,1080,150,15,1},
  {177147,73728,15435,2160,225,18,1},
  {3000000,1240029,258048,36015,3780,315,21,1},..
		

Links

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

Crossrefs

Cf. A089464(row sums), A089465(diagonal), A089460, A088956.

Programs

Mathematica

Flatten[Table[3(n-k+3)^(n-k-1) Binomial[n,k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jun 26 2013 *)

PARI

for(n=0,10, for(k=0,n, print1(3*(n-k+3)^(n-k-1)*binomial(n,k), ", "))) \\ G. C. Greubel, Nov 17 2017

Formula

T(n, k) = 3*(n-k+3)^(n-k-1)*C(n, k).

E.g.f.: exp(x*y)*(-LambertW(-y)/y)^3.

Note: (-LambertW(-y)/y)^3 = Sum_{n>=0} 3*(n+3)^(n-1)*y^n/n!.