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

%I A089460 #11 Apr 05 2025 19:53:41
%S A089460 1,2,1,8,4,1,50,24,6,1,432,200,48,8,1,4802,2160,500,80,10,1,65536,
%T A089460 28812,6480,1000,120,12,1,1062882,458752,100842,15120,1750,168,14,1,
%U A089460 20000000,8503056,1835008,268912,30240,2800,224,16,1,428717762,180000000,38263752,5505024,605052,54432,4200,288,18,1
%N A089460 Triangle, read by rows, of coefficients for the second iteration of the hyperbinomial transform.
%C A089460 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 A089460 G. C. Greubel, <a href="/A089460/b089460.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A089460 T(n, k) = 2*(n-k+2)^(n-k-1)*C(n, k).
%F A089460 E.g.f.: exp(x*y)*(-LambertW(-y)/y)^2.
%F A089460 Note: (-LambertW(-y)/y)^2 = Sum_{n>=0} 2*(n+2)^(n-1)*y^n/n!.
%e A089460 Rows begin:
%e A089460   {1},
%e A089460   {2,1},
%e A089460   {8,4,1},
%e A089460   {50,24,6,1},
%e A089460   {432,200,48,8,1},
%e A089460   {4802,2160,500,80,10,1},
%e A089460   {65536,28812,6480,1000,120,12,1},
%e A089460   {1062882,458752,100842,15120,1750,168,14,1},..
%t A089460 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 A089460 (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 A089460 Cf. A089461(row sums), A089462(diagonal), A089463, A088956.
%K A089460 nonn,tabl
%O A089460 0,2
%A A089460 _Paul D. Hanna_, Nov 05 2003