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A139167
Triangle T(n,k) read by rows: the coefficient [x^k] of the polynomial (n-1)! *sum_{i=0..n} Fibonacci(i)*binomial(x,n-i), read by rows, 0<=k<n.
2
1, 1, 1, 4, 1, 1, 18, 11, 0, 1, 120, 50, 23, -2, 1, 960, 494, 65, 45, -5, 1, 9360, 4344, 1354, -15, 85, -9, 1, 105840, 51876, 10444, 3409, -350, 154, -14, 1, 1370880, 653232, 172444, 13300, 8729, -1232, 266, -20, 1, 19958400, 9654480, 2194380, 483272, -13923, 22449, -3150, 438, -27, 1
OFFSET
1,4
COMMENTS
Row sums are 1, 2, 6, 30, 192, 1560, 15120, 171360, 2217600, 32296320,... (see A078700)
REFERENCES
Brendan Hassett, Introduction to algebraic Geometry,Cambridge University Press. New York,2007, page 229
EXAMPLE
1;
1, 1;
4, 1, 1;
18, 11, 0, 1;
120, 50, 23, -2, 1;
960, 494, 65, 45, -5, 1;
9360, 4344, 1354, -15,85, -9, 1;
105840, 51876, 10444, 3409, -350, 154, -14, 1;
1370880, 653232, 172444, 13300, 8729, -1232, 266, -20, 1;
19958400, 9654480, 2194380, 483272, -13923, 22449, -3150, 438, -27, 1;
MAPLE
B := proc(x, k)
mul( (x-i+1)/i, i=1..k) ;
end proc:
A139167 := proc(n, k)
local f, i ;
f := 0 ;
for i from 0 to n do
f := f+combinat[fibonacci](i)*B(x, n-i) ;
end do;
%*(n-1)! ;
coeftayl(%, x=0, k) ;
end proc: # R. J. Mathar, May 08 2013
MATHEMATICA
Clear[a, p, x] a[0] = 0; a[1] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2]; p[x, 0] = a[0]; p[x_, n_] := p[x, n] = Sum[a[i]*Binomial[x, n - i], {i, 0, n}]; Table[If[n > 0, ExpandAll[(n - 1)!*p[x, n]], 0], {n, 0, 10}]; a = Table[CoefficientList[If[n > 0, ExpandAll[(n - 1)!*p[x, n]], 0], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[If[n > 0, ExpandAll[(n - 1)!*p[x, n]], 0], x]], {n, 0, 10}]
CROSSREFS
Cf. A000045.
Sequence in context: A034802 A177262 A203092 * A211709 A323849 A254442
KEYWORD
tabl,sign
AUTHOR
Roger L. Bagula, Jun 05 2008
EXTENSIONS
Edited by R. J. Mathar, May 08 2013
STATUS
approved