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Triangle read by rows: T(n,k)=(-1)^k*(2n/(2n-k))5^(n-k)*binomial(2n-k,k) (0<=k<=n, n>=1).
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%I #11 Jan 05 2025 19:51:38

%S 5,-2,25,-20,2,125,-150,45,-2,625,-1000,500,-80,2,3125,-6250,4375,

%T -1250,125,-2,15625,-37500,33750,-14000,2625,-180,2,78125,-218750,

%U 240625,-131250,36750,-4900,245,-2,390625,-1250000,1625000,-1100000,412500,-84000,8400,-320,2

%N Triangle read by rows: T(n,k)=(-1)^k*(2n/(2n-k))5^(n-k)*binomial(2n-k,k) (0<=k<=n, n>=1).

%C Row n contains n+1 terms. Row sums yield the even-subscripted Lucas numbers (A005248).

%H P. Filipponi, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/30-3/filipponi.pdf">Waring's formula, the binomial formula and generalized Fibonacci matrices</a>, The Fibonacci Quarterly, 30, 1992, 225-231.

%H P. Filipponi, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/36-1/filipponi1.pdf">Combinatorial expressions for Lucas numbers</a>, The Fibonacci Quarterly, 36, 1998, 63-65.

%e Triangle starts:

%e 5, -2;

%e 25, -20, 2;

%e 125, -150, 45, -2;

%e 625, -1000, 500, -80, 2; - _Michel Marcus_, Apr 09 2013

%p T:=proc(n,k) if k<=n then (-1)^k*2*n*5^(n-k)*binomial(2*n-k,k)/(2*n-k) else 0 fi end: for n from 1 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

%Y Cf. A005248.

%K sign,tabf

%O 1,1

%A _Emeric Deutsch_, Mar 02 2005