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A244119
Triangle read by rows: terms of a binomial decomposition of 1 as Sum(k=0..n)T(n,k).
30
1, 0, 1, 0, -2, 3, 0, 3, -18, 16, 0, -4, 72, -192, 125, 0, 5, -240, 1440, -2500, 1296, 0, -6, 720, -8640, 30000, -38880, 16807, 0, 7, -2016, 45360, -280000, 680400, -705894, 262144, 0, -8, 5376, -217728, 2240000, -9072000, 16941456, -14680064, 4782969
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
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);
KEYWORD
sign,tabl
AUTHOR
Stanislav Sykora, Jun 21 2014
STATUS
approved