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A144457
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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}]].
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0
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1, -1, 1, -3, -8, 3, -15, -119, -217, 15, -105, -1574, -7440, -10954, 105, -945, -22679, -194646, -702874, -892281, 945, -10395, -363824, -4885615, -31288480, -94892945, -108046896, 10395, -135135, -6486479, -124999827, -1232430275, -6521470845, -17442096461, -18261339153, 135135
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OFFSET
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1,4
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COMMENTS
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The name contains an unmatched parenthesis. - Editors, Mar 13 2024
Row sums are:
{1, 0, -8, -336, -19968, -1812480, -239477760, -43588823040, -10461389783040, -3201186759966720, -1216451002230374400}.
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LINKS
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FORMULA
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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)).
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EXAMPLE
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{1},
{-1, 1},
{-3, -8, 3},
{-15, -119, -217, 15},
{-105, -1574, -7440, -10954,105},
{-945, -22679, -194646, -702874, -892281,945},
{-10395, -363824, -4885615, -31288480, -94892945, -108046896, 10395},
{-135135, -6486479, -124999827, -1232430275, -6521470845, -17442096461, -18261339153, 135135}
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MATHEMATICA
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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[%]
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CROSSREFS
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KEYWORD
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sign,uned
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AUTHOR
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STATUS
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approved
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