A101033 Triangle read by rows giving the coefficients of general sum formulas of n-th Lucas numbers (A000204). The k-th row (k>=1) contains T(i,k) for i=1 to 2*k-1, where T(i,k) satisfies L(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, -2, -3, 2, 15, 51, 65, 27, 6, -148, -945, -2292, -2776, -1680, -405, 24, 2290, 19580, 71965, 145525, 175244, 125950, 50085, 8505, 120, -41676, -473072, -2340400, -6676835, -12132890, -14587261, -11619692, -5290005, -1752030, -229635, 720, 943908, 13132532, 81977672, 303352938, 740797855

Offset

1,3

Links

Table of n, a(n) for n=1..42.

Ron Knott, The Lucas Numbers in Pascal's Triangle.

Example

L(7)= (1/(7-1)!) * [ 7^(7-1) -{-1+2*(7-2)+3*C(7-2,2)}*7^(7-2) +{2+15*(7-3)+51*C(7-3,2)+65*C(7-3,3) +27*C(7-3,4)}*7^(7-3) -{-6+148*(7-4)+945*C(7-4,2)+2292*C(7-4,3)}*7^(7-4) +... ]
= (1/6!) * [ 7^6 -{-1+10+30}*7^5 +{2+60+306+260+27}*7^4 -{-6+444+2835+2292}*7^3 +{24+4580+19580}*7^2 -{-120+41676}*7 +{720} ] = (1/6!) * [ 7^6 -39*7^5 +655*7^4 -5565*7^3 +24184*7^2 -41556*7 +720 ]
= (1/720) * [ 117649 -655473 +1572655 -1908795 +1185016 -290892 +720 ] = 20880/720 = 29.

Crossrefs

Cf. A101032, A000204, A100492, A099731, A000045, A094216, A094638, A000108.

Sequence in context: A367566 A164661 A104507 * A136454 A025522 A350622

Adjacent sequences: A101030 A101031 A101032 * A101034 A101035 A101036

Keyword

easy,sign,tabl

Author

André F. Labossière, Nov 30 2004

Status

approved