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A178640
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Triangle T(n,k) with the coefficient [x^k] of the series (-1)^(n+1) * (x-1)^(n+1) * Sum_{j>=0} (5+8*j)^n*x^j in row n, column k.
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1
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1, 5, 3, 25, 94, 9, 125, 1697, 1223, 27, 625, 25436, 57926, 14236, 81, 3125, 352543, 1903218, 1513438, 159593, 243, 15625, 4717434, 52306583, 95276588, 34660263, 1766458, 729, 78125, 62123517, 1301287905, 4593751457, 3854897607, 738035607
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OFFSET
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0,2
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COMMENTS
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Row sums are 1, 8, 128, 3072, 98304, 3932160, 188743680, 10569646080, 676457349120, 48704929136640, 3896394330931200, ....
Other pairs of consecutive Fibonacci numbers could be substituted for the two main parameters (5,8) in the definition. Using the pair (0,1) or (1,1) yields A008292. Using the pair (1,2) yields A060187.
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LINKS
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FORMULA
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EXAMPLE
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1;
5, 3;
25, 94, 9;
125, 1697, 1223, 27;
625, 25436, 57926, 14236, 81;
3125, 352543, 1903218, 1513438, 159593, 243;
15625, 4717434, 52306583, 95276588, 34660263, 1766458, 729;
78125, 62123517, 1301287905, 4593751457, 3854897607, 738035607, 19469675, 2187;
390625, 812215096, 30495345372, 189174172168, 303412512454, 137293837704, 15054569308, 214299832, 6561;
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MAPLE
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A178640 := proc(n, k) (-1)^(n+1)*(x-1)^(n+1)*add( (5+8*j)^n*x^j, j=0..k) ; coeftayl(%, x=0, k) ; end proc: # R. J. Mathar, Apr 05 2011
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MATHEMATICA
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Clear[m, m0, t, n, k]
m0 = {{1, 1}, {1, 0}}
m[l_] := MatrixPower[m0, l]
t[l_, k_] = If[l == 0, 1, m[l][[1, 1]]*k + m[l][[1, 2]]]
p[x_, n_, l_] := (-1)^(n + 1)*(-1 + x)^(n + 1)*Sum[t[l, k]^ n*x^k, {k, 0, Infinity}]
Table[Flatten[Table[CoefficientList[FullSimplify[ExpandAll[p[x, n, l]]], x], {n, 0, 10}]], {l, 0, 10}]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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