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A089460 Triangle, read by rows, of coefficients for the second iteration of the hyperbinomial transform. 4

%I #8 Nov 18 2017 04:22:47

%S 1,2,1,8,4,1,50,24,6,1,432,200,48,8,1,4802,2160,500,80,10,1,65536,

%T 28812,6480,1000,120,12,1,1062882,458752,100842,15120,1750,168,14,1,

%U 20000000,8503056,1835008,268912,30240,2800,224,16,1,428717762,180000000,38263752,5505024,605052,54432,4200,288,18,1

%N Triangle, read by rows, of coefficients for the second iteration of the hyperbinomial transform.

%C 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.

%H G. C. Greubel, <a href="/A089460/b089460.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F T(n, k) = 2*(n-k+2)^(n-k-1)*C(n, k).

%F E.g.f.: exp(x*y)*(-LambertW(-y)/y)^2.

%F Note: (-LambertW(-y)/y)^2 = sum(n>=0, 2*(n+2)^(n-1)*y^n/n!).

%e Rows begin:

%e {1},

%e {2,1},

%e {8,4,1},

%e {50,24,6,1},

%e {432,200,48,8,1},

%e {4802,2160,500,80,10,1},

%e {65536,28812,6480,1000,120,12,1},

%e {1062882,458752,100842,15120,1750,168,14,1},..

%t 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 *)

%o (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

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

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Nov 05 2003

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)