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A244127
Triangle read by rows: terms T(n,k) of a binomial decomposition of 2^n-1 as Sum(k=0..n)T(n,k).
28
0, 0, 1, 0, 0, 3, 0, 0, -9, 16, 0, 0, 18, -128, 125, 0, 0, -30, 640, -1875, 1296, 0, 0, 45, -2560, 16875, -31104, 16807, 0, 0, -63, 8960, -118125, 435456, -588245, 262144, 0, 0, 84, -28672, 708750, -4644864, 11764900, -12582912, 4782969
OFFSET
0,6
COMMENTS
T(n,k)=(1+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.(6), with b=-1 and a=1.
EXAMPLE
First rows of the triangle, all summing up to 2^n-1:
0,
0, 1,
0, 0, 3,
0, 0, -9, 16,
0, 0, 18, -128, 125,
0, 0, -30, 640, -1875, 1296,
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]=(1-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 21 2014
STATUS
approved