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Search: id:A089460
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| A089460 |
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Triangle, read by rows, of coefficients for the second iteration of the hyperbinomial transform. |
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+0
4
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| 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
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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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.
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FORMULA
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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!).
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EXAMPLE
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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},..
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CROSSREFS
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Cf. A089461(row sums), A089462(diagonal), A089463, A088956.
Sequence in context: A110171 A104988 A136225 this_sequence A135520 A136230 A004732
Adjacent sequences: A089457 A089458 A089459 this_sequence A089461 A089462 A089463
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Nov 05 2003
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