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A144406
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Polynomial expansion as anti-diagonal of: p(x,n)=(x-1)/(x^n*(-x+(2*x-1)/x^n). Based on the Pisot general polynomial type q(x,n)=x^n-(x^n-1)/(x-1).
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0
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1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4, 5, 1, 1, 1, 2, 4, 7, 8, 1, 1, 1, 2, 4, 8, 13, 13, 1, 1, 1, 2, 4, 8, 15, 24, 21, 1, 1, 1, 2, 4, 8, 16, 29, 44, 34, 1, 1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1, 1, 1, 2, 4, 8, 16, 32, 61, 108, 149, 89, 1, 1, 1, 2, 4, 8, 16, 32, 63, 120, 208, 274
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OFFSET
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1,9
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COMMENTS
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Row sums are:
{1, 2, 3, 5, 8, 14, 24, 43, 77, 140, 256, 472, 874, 1628, 3045}.
(Start) From L. Edson Jeffery, Nov 27, 2011: Sequence can also be read from the anti-diagonals of the table:
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}
{1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...}
{1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, ...}
{1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, ...}
{1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, ...}
{1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, ...}
{1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, ...}
{1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, ...}
{1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, ...}
etc.. Conjectures: (i) Sequence in row k has generating function
F_k(x)=1/(1-x-x^2-...-x^k), k in {1,2,...}; (ii) Assuming the sequences successively tend to A000079, the absolute values of non-zero differences between two successive row sequences tend to A045623={1,2,5,12,28,64,144,320,704,1536,...}, as k -> infinity. (End)
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LINKS
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Table of n, a(n) for n=1..89.
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FORMULA
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p(x,n)=(x-1)/(x^n*(-x+(2*x-1)/x^n);t(n,m)=anti_diagonal_expansion(p(x,n)).
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EXAMPLE
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{1},
{1, 1},
{1, 1, 1},
{1, 1, 2, 1},
{1, 1, 2, 3, 1},
{1, 1, 2, 4, 5, 1},
{1, 1, 2, 4, 7, 8, 1},
{1, 1, 2, 4, 8, 13, 13, 1},
{1, 1, 2, 4, 8, 15, 24, 21, 1},
{1, 1, 2, 4, 8, 16, 29, 44, 34, 1},
{1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1},
{1, 1, 2, 4, 8, 16, 32, 61, 108, 149, 89, 1},
{1, 1, 2, 4, 8, 16, 32, 63, 120, 208, 274, 144, 1},
{1, 1, 2, 4, 8, 16, 32, 64, 125, 236, 401, 504, 233, 1},
{1, 1, 2, 4, 8, 16, 32, 64, 127, 248, 464, 773, 927, 377, 1}
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MATHEMATICA
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Clear[f, b, a, g, h, n, t]; g[x_, n_] = x^(n) - (x^n - 1)/(x - 1); h[x_, n_] = FullSimplify[ExpandAll[x^(n)*g[1/x, n]]]; f[t_, n_] := 1/h[t, n]; Series[f[t, m], {t, 0, 30}], n], {n, 0, 30}], {m, 1, 31}]; b = Table[Table[a[[n - m + 1]][[m]], {m, 1, n }], {n, 1, 15}]; Flatten[b]
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CROSSREFS
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Cf. A000079, A045623.
Sequence in context: A205573 A119338 A054124 * A179748 A096670 A130461
Adjacent sequences: A144403 A144404 A144405 * A144407 A144408 A144409
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KEYWORD
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nonn,uned
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AUTHOR
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Roger L. Bagula and Gary W. Adamson, Sep 29 2008
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STATUS
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approved
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