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%I #6 Jun 16 2016 23:27:26
%S 1,-1,-4,-3,10,49,95,83,27,-90,-740,-2415,-4110,-3890,-1950,-405,1320,
%T 14054,64116,164059,258461,257604,159070,55755,8505,-23640,-318684,
%U -1881532,-6452300,-14294605,-21442540,-22106669,-15496012,-7078575,-1905120,-229635,523440,8474100,61424596
%N Triangle read by rows giving the coefficients of general sum formulas of n-th Fibonacci numbers (A000045). The k-th row (k>=1) contains T(i,k) for i=1 to 2*k-1, where T(i,k) satisfies F(n) = Sum_{k=1..n} Sum_{i=1..2*k-1} T(i,k) * C(n-k,i-1) * n^(n-k) / (n-1)!.
%e F(7) = (1/(7-1)!) * [ 7^(7-1) -{1+4*(7-2)+3*C(7-2,2)}*7^(7-2) +{10+49*(7-3)+95*C(7-3,2)+83*C(7-3,3) +27*C(7-3,4)}*7^(7-3) -{90+740*(7-4)+2415*C(7-4,2)+4110*C(7-4,3)}*7^(7-4) +... ]
%e = (1/6!) * [ 7^6 -{1+20+30}*7^5 +{10+196+570+332+27}*7^4 -{90+2220+7245+4110}*7^3 +{1320+28108 +64116}*7^2 -{23640+318684}*7 +{523440} ]
%e = (1/6!) * [ 7^6 -51*7^5 +1135*7^4 -13665*7^3 +93544*7^2 -342324*7 +523440 ]
%e = (1/720) * [ 117649 -857157 +2725135 -4687095 +4583656 -2396268 +523440 ] = 9360/720 = 13.
%Y Cf. A099731, A000045, A094216, A094638, A000108.
%K easy,sign,tabl
%O 1,3
%A _André F. Labossière_, Nov 22 2004
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