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A178640
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.
1
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
OFFSET
0,2
COMMENTS
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.
FORMULA
T(n,0) = 5^n = A000351(n).
EXAMPLE
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;
MAPLE
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
MATHEMATICA
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}]
CROSSREFS
Sequence in context: A032532 A111108 A038245 * A273153 A288812 A288898
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, May 31 2010
STATUS
approved