OFFSET
0,2
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.
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
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!).
EXAMPLE
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},..
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Nov 05 2003
STATUS
approved