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A244133
Triangle read by rows: terms T(n,k) of a binomial decomposition of n as Sum(k=0..n)T(n,k).
28
0, 0, 1, 0, 0, 2, 0, 0, -6, 9, 0, 0, 12, -72, 64, 0, 0, -20, 360, -960, 625, 0, 0, 30, -1440, 8640, -15000, 7776, 0, 0, -42, 5040, -60480, 210000, -272160, 117649, 0, 0, 56, -16128, 362880, -2240000, 5443200, -5647152, 2097152, 0, 0, -72, 48384, -1959552, 20160000, -81648000, 152473104, -132120576, 43046721
OFFSET
0,6
COMMENTS
T(n,k)=(k)^(k-1)*(1-k)^(n-k)*binomial(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
First rows of the triangle, all summing up to n:
0,
0, 1,
0, 0, 2,
0, 0, -6, 9,
0, 0, 12, -72, 64,
0, 0, -20, 360, -960, 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)*binomial(n, k); ); );
return(v); }
a=seq(100, -1);
KEYWORD
sign,tabl
AUTHOR
Stanislav Sykora, Jun 22 2014
STATUS
approved