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A089460 Triangle, read by rows, of coefficients for the second iteration of the hyperbinomial transform. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified June 18 02:37 EDT 2021. Contains 345098 sequences. (Running on oeis4.)