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A244130
Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n as Sum_{k=0..n} T(n,k)*binomial(n,k).
28
0, 0, 1, 0, 2, -2, 0, 4, -6, 9, 0, 8, -18, 36, -64, 0, 16, -54, 144, -320, 625, 0, 32, -162, 576, -1600, 3750, -7776, 0, 64, -486, 2304, -8000, 22500, -54432, 117649, 0, 128, -1458, 9216, -40000, 135000, -381024, 941192, -2097152, 0, 256, -4374, 36864, -200000, 810000, -2667168, 7529536, -18874368, 43046721
OFFSET
0,5
COMMENTS
T(n,k) = (-k)^(k-1)*(1+k)^(n-k) for k>0, while T(n,0)=0 by convention.
LINKS
S. Sykora, An Abel's Identity and its Corollaries, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(12), with b=1.
EXAMPLE
The first rows of the triangle are:
0,
0, 1,
0, 2, -2,
0, 4, -6, 9,
0, 8, -18, 36, -64,
0, 16, -54, 144, -320, 625,
PROG
(PARI) seq(nmax, b)={my(v, n, k, irow);
v = vector((nmax+1)*(nmax+2)/2); v[1]=0;
for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
for(k=1, n, v[irow+k]=(-k*b)^(k-1)*(1+k*b)^(n-k); ); );
return(v); }
a=seq(100, 1);
KEYWORD
sign,tabl
AUTHOR
Stanislav Sykora, Jun 22 2014
STATUS
approved