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A244124
Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 2^n-1 as Sum(k=0..n)T(n,k)*binomial(n,k).
28
0, 0, 1, 0, 2, -1, 0, 4, -3, 4, 0, 8, -9, 16, -27, 0, 16, -27, 64, -135, 256, 0, 32, -81, 256, -675, 1536, -3125, 0, 64, -243, 1024, -3375, 9216, -21875, 46656, 0, 128, -729, 4096, -16875, 55296, -153125, 373248, -823543, 0, 256, -2187, 16384, -84375, 331776, -1071875, 2985984, -7411887, 16777216
OFFSET
0,5
COMMENTS
T(n,k)=(1-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.(6), with b=1 and a=1.
EXAMPLE
The first rows of the triangle are:
0
0 1
0 2 -1
0 4 -3 4
0 8 -9 16 -27
0 16 -27 64 -135 256
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); ); );
return(v); }
a=seq(100, 1)
KEYWORD
sign,tabl
AUTHOR
Stanislav Sykora, Jun 21 2014
STATUS
approved