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

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

Author

Roger L. Bagula, Jun 05 2008

Keywords

• tabl
• sign

Comments

Row sums are 1, 2, 6, 30, 192, 1560, 15120, 171360, 2217600, 32296320,... (see A078700)

Examples

			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;
		

References

• Brendan Hassett, Introduction to algebraic Geometry,Cambridge University Press. New York,2007, page 229

Crossrefs

Cf. A000045.

Programs

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}]

Extensions

Edited by R. J. Mathar, May 08 2013