A255811 Rectangular array: row n gives the numerators in the positive convolutory n-th root of (1,1,1,...).
1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 2, 1, 1, 1, 35, 14, 5, 1, 1, 1, 63, 35, 15, 3, 1, 1, 1, 231, 91, 195, 11, 7, 1, 1, 1, 429, 728, 663, 44, 91, 4, 1, 1, 1, 6435, 1976, 4641, 924, 1729, 20, 9, 1, 1, 1, 12155, 5434, 16575, 4004, 8645, 110, 51, 5, 1, 1, 1
Offset: 1
Examples
First, regarding the numbers numerator/denominator, we have row 1: 1,1,1,1,1,1,1,1,1,1,1,1,1,..., the 0th self-convolution of (1,1,1,...); row 2: 1,1/2,3/8,5/16,35/128,63/256, ..., convolutory sqrt of (1,1,1,...); row 3: 1,1/3,2/9,14/81,35/243,91/729,..., convolutory 3rd root; row 4: 1,1/4,5/32,15/128,195/2048,663/8192,..., convolutoary 4th root. Taking only numerators: row 1: 1,1,1,1,1,1,1,... row 2: 1,1,3,5,35,63,... row 3: 1,1,2,14,35,91,... row 4: 1,1,5,15,195,663,...
Links
- Clark Kimberling, Antidiagonals n = 1..60, flattened
Programs
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Mathematica
z = 15; t[n_] := CoefficientList[Normal[Series[(1 - t)^(-1/n), {t, 0, z}]], t]; u = Table[Numerator[t[n]], {n, 1, z}] TableForm[Table[u[[n, k]], {n, 1, z}, {k, 1, z}]] (* A255811 array *) Table[u[[n - k + 1, k]], {n, z}, {k, n, 1, -1}] // Flatten (* A255811 sequence *) v = Table[Denominator[t[n]], {n, 1, z}] TableForm[Table[v[[n, k]], {n, 1, z}, {k, 1, z}]] (* A255812 array *) Table[v[[n - k + 1, k]], {n, z}, {k, n, 1, -1}] // Flatten (* A255812 sequence *)
Formula
G.f. of s: (1 - t)^(-1/n).
Comments