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A144457 Coefficients of polynomials based on the generalized factorial at k=2 (A001147): b(n)=b(n-1+k; a(n)=b(n)*a(n-1); p(x,n)=If[n == 0, 1, a(n - 1)*(x - a(n - 1))*Product[x + 1/b(i), {i, 1, n - 1}]].

%I A144457 #5 Mar 13 2024 19:21:13
%S A144457 1,-1,1,-3,-8,3,-15,-119,-217,15,-105,-1574,-7440,-10954,105,-945,
%T A144457 -22679,-194646,-702874,-892281,945,-10395,-363824,-4885615,-31288480,
%U A144457 -94892945,-108046896,10395,-135135,-6486479,-124999827,-1232430275,-6521470845,-17442096461,-18261339153,135135
%N A144457 Coefficients of polynomials based on the generalized factorial at k=2 (A001147): b(n)=b(n-1+k; a(n)=b(n)*a(n-1); p(x,n)=If[n == 0, 1, a(n - 1)*(x - a(n - 1))*Product[x + 1/b(i), {i, 1, n - 1}]].
%C A144457 The name contains an unmatched parenthesis. - Editors, Mar 13 2024
%C A144457 Row sums are:
%C A144457 {1, 0, -8, -336, -19968, -1812480, -239477760, -43588823040, -10461389783040, -3201186759966720, -1216451002230374400}.
%F A144457 b(n)=b(n-1+k; a(n)=b(n)*a(n-1); p(x,n)=If[n == 0, 1, a(n - 1)*(x - a(n - 1))*Product[x + 1/b(i), {i, 1, n - 1}]]; t(n,m)=coefficients(p(x,n)).
%e A144457 {1},
%e A144457 {-1, 1},
%e A144457 {-3, -8, 3},
%e A144457 {-15, -119, -217, 15},
%e A144457 {-105, -1574, -7440, -10954,105},
%e A144457 {-945, -22679, -194646, -702874, -892281,945},
%e A144457 {-10395, -363824, -4885615, -31288480, -94892945, -108046896, 10395},
%e A144457 {-135135, -6486479, -124999827, -1232430275, -6521470845, -17442096461, -18261339153, 135135}
%t A144457 Clear[a, b, p, x, n]; k = 2; b[0] = 1; b[n_] := b[n] = b[n - 1] + k; a[0] = 1; a[n_] := a[n] = b[n]*a[n - 1]; p[x_, n_] = If[n == 0, 1, a[n - 1]*(x - a[n - 1])*Product[x + 1/b[i], {i, 1, n - 1}]]; Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[%]
%Y A144457 Cf. A001147.
%K A144457 sign,uned
%O A144457 1,4
%A A144457 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 07 2008