OFFSET
0,5
COMMENTS
T(n,k)=(1+k)^(k-1)*(-k)^(n-k)*binomial(n,k) for k>0, while T(n,0)=0^n by convention.
Sequence A161628, arising from a different context, appears to be the same, but with opposite signs of odd rows.
LINKS
Stanislav Sykora, Table of n, a(n) for rows 0..100
S. Sykora, An Abel's Identity and its Corollaries, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(4) with b=-1.
EXAMPLE
First rows of the triangle, all summing up to 1:
1
0 1
0 -2 3
0 3 -18 16
0 -4 72 -192 125
0 5 -240 1440 -2500 1296
MAPLE
A244119 := (n, k) -> (1+k)^(k-1)*(-k)^(n-k)*binomial(n, k):
seq(seq(A244119(n, k), k = 0..n), n = 0..8); # Peter Luschny, Jan 29 2023
PROG
(PARI) seq(nmax, b)={my(v, n, k, irow);
v = vector((nmax+1)*(nmax+2)/2); v[1]=1;
for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
for(k=1, n, v[irow+k]=(1-k*b)^(k-1)*(k*b)^(n-k)*binomial(n, k); );
); return(v); }
a=seq(100, -1);
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Stanislav Sykora, Jun 21 2014
STATUS
approved