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

0,2

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.

Links

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

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!).

Example

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},..

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A343296 A136225 A341724 * A308695 A278111 A223550

Adjacent sequences: A089457 A089458 A089459 * A089461 A089462 A089463

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 05 2003

Status

approved