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A089463 Triangle, read by rows, of coefficients for the third iteration of the hyperbinomial transform.

%I A089463 #13 Apr 05 2025 19:53:31
%S A089463 1,3,1,15,6,1,108,45,9,1,1029,432,90,12,1,12288,5145,1080,150,15,1,
%T A089463 177147,73728,15435,2160,225,18,1,3000000,1240029,258048,36015,3780,
%U A089463 315,21,1,58461513,24000000,4960116,688128,72030,6048,420,24,1,1289945088
%N A089463 Triangle, read by rows, of coefficients for the third iteration of the hyperbinomial transform.
%C A089463 Equals the matrix cube of A088956 when treated as a lower triangular matrix. The 3rd 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) = 3*(n-k+3)^(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 3rd hyperbinomial transform of any diagonal results in the 3rd diagonal lower in the table.
%H A089463 G. C. Greubel, <a href="/A089463/b089463.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A089463 T(n, k) = 3*(n-k+3)^(n-k-1)*C(n, k).
%F A089463 E.g.f.: exp(x*y)*(-LambertW(-y)/y)^3.
%F A089463 Note: (-LambertW(-y)/y)^3 = Sum_{n>=0} 3*(n+3)^(n-1)*y^n/n!.
%e A089463 Rows begin:
%e A089463   {1},
%e A089463   {3,1},
%e A089463   {15,6,1},
%e A089463   {108,45,9,1},
%e A089463   {1029,432,90,12,1},
%e A089463   {12288,5145,1080,150,15,1},
%e A089463   {177147,73728,15435,2160,225,18,1},
%e A089463   {3000000,1240029,258048,36015,3780,315,21,1},..
%t A089463 Flatten[Table[3(n-k+3)^(n-k-1) Binomial[n,k],{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, Jun 26 2013 *)
%o A089463 (PARI) for(n=0,10, for(k=0,n, print1(3*(n-k+3)^(n-k-1)*binomial(n,k), ", "))) \\ _G. C. Greubel_, Nov 17 2017
%Y A089463 Cf. A089464(row sums), A089465(diagonal), A089460, A088956.
%K A089463 nonn,tabl
%O A089463 0,2
%A A089463 _Paul D. Hanna_, Nov 05 2003