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.

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

Links

Table of n, a(n) for n=1..55.

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

Adjacent sequences: A139164 A139165 A139166 * A139168 A139169 A139170

Keyword

tabl,sign

Author

Roger L. Bagula, Jun 05 2008

Extensions

Edited by R. J. Mathar, May 08 2013

Status

approved