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A168296 Worpitzky form polynomials for the {1,16,1} A142462 sequence: p(x,n) = Sum_{k=1..n} A(n, k)*binomial(x + k - 1, n - 1).

%I A168296 #17 Feb 07 2025 05:37:25
%S A168296 1,1,2,2,18,18,6,156,432,288,24,792,7416,13248,6624,120,-11280,64800,
%T A168296 374400,496800,198720,720,-62640,-1254960,4968000,20865600,22057920,
%U A168296 7352640,5040,24012000,-11854080,-125677440,389491200,1288103040,1132306560,323516160
%N A168296 Worpitzky form polynomials for the {1,16,1} A142462 sequence: p(x,n) = Sum_{k=1..n} A(n, k)*binomial(x + k - 1, n - 1).
%F A168296 p(x,n) = Sum_{k=1..n} A(n, k)*binomial(x + k - 1, n - 1).
%e A168296 Triangle begins:
%e A168296   {1},
%e A168296   {1, 2},
%e A168296   {2, 18, 18},
%e A168296   {6, 156, 432, 288},
%e A168296   {24, 792, 7416, 13248, 6624},
%e A168296   {120, -11280, 64800, 374400, 496800, 198720},
%e A168296   {720, -62640, -1254960, 4968000, 20865600, 22057920, 7352640},
%e A168296   {5040, 24012000, -11854080, -125677440, 389491200, 1288103040, 1132306560, 323516160},
%e A168296   ...
%t A168296 (* Worpitzky form polynomials for A142462 *)
%t A168296 m = 7;
%t A168296 A[n_, 1] := 1 A[n_, n_] := 1
%t A168296 A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k];
%t A168296 a = Table[A[n, k], {n, 10}, {k, n}];
%t A168296 p[x_, n_] = Sum[a[[n, k]]*Binomial[x + k - 1, n - 1], {k, 1, n}];
%t A168296 Table[CoefficientList[Expand[(n - 1)!*p[x, n]], x], {n, 1, 10}];
%t A168296 Flatten[%]
%Y A168296 Cf. A142462.
%K A168296 uned,sign,tabl
%O A168296 1,3
%A A168296 _Roger L. Bagula_, Nov 22 2009