A100492 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)!.

1, -1, -4, -3, 10, 49, 95, 83, 27, -90, -740, -2415, -4110, -3890, -1950, -405, 1320, 14054, 64116, 164059, 258461, 257604, 159070, 55755, 8505, -23640, -318684, -1881532, -6452300, -14294605, -21442540, -22106669, -15496012, -7078575, -1905120, -229635, 523440, 8474100, 61424596

Offset

1,3

Links

Table of n, a(n) for n=1..39.

Example

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) +... ]
= (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} ]
= (1/6!) * [ 7^6 -51*7^5 +1135*7^4 -13665*7^3 +93544*7^2 -342324*7 +523440 ]
= (1/720) * [ 117649 -857157 +2725135 -4687095 +4583656 -2396268 +523440 ] = 9360/720 = 13.

Crossrefs

Cf. A099731, A000045, A094216, A094638, A000108.

Sequence in context: A242391 A241862 A222510 * A072183 A353341 A346614

Adjacent sequences: A100489 A100490 A100491 * A100493 A100494 A100495

Keyword

easy,sign,tabl

Author

André F. Labossière, Nov 22 2004

Status

approved