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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

%I A139167 #9 May 08 2013 13:55:36
%S A139167 1,1,1,4,1,1,18,11,0,1,120,50,23,-2,1,960,494,65,45,-5,1,9360,4344,
%T A139167 1354,-15,85,-9,1,105840,51876,10444,3409,-350,154,-14,1,1370880,
%U A139167 653232,172444,13300,8729,-1232,266,-20,1,19958400,9654480,2194380,483272,-13923,22449,-3150,438,-27,1
%N 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.
%C A139167 Row sums are 1, 2, 6, 30, 192, 1560, 15120, 171360, 2217600, 32296320,... (see A078700)
%D A139167 Brendan Hassett, Introduction to algebraic Geometry,Cambridge University Press. New York,2007, page 229
%e A139167 1;
%e A139167 1, 1;
%e A139167 4, 1, 1;
%e A139167 18, 11, 0, 1;
%e A139167 120, 50, 23, -2, 1;
%e A139167 960, 494, 65, 45, -5, 1;
%e A139167 9360, 4344, 1354, -15,85, -9, 1;
%e A139167 105840, 51876, 10444, 3409, -350, 154, -14, 1;
%e A139167 1370880, 653232, 172444, 13300, 8729, -1232, 266, -20, 1;
%e A139167 19958400, 9654480, 2194380, 483272, -13923, 22449, -3150, 438, -27, 1;
%p A139167 B := proc(x,k)
%p A139167     mul( (x-i+1)/i,i=1..k) ;
%p A139167 end proc:
%p A139167 A139167 := proc(n,k)
%p A139167     local f,i ;
%p A139167     f := 0 ;
%p A139167     for i from 0 to n do
%p A139167         f := f+combinat[fibonacci](i)*B(x,n-i) ;
%p A139167     end do;
%p A139167     %*(n-1)! ;
%p A139167     coeftayl(%,x=0,k) ;
%p A139167 end proc: # _R. J. Mathar_, May 08 2013
%t A139167 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}]
%Y A139167 Cf. A000045.
%K A139167 tabl,sign
%O A139167 1,4
%A A139167 _Roger L. Bagula_, Jun 05 2008
%E A139167 Edited by _R. J. Mathar_, May 08 2013