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A244140
Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n*(-1)^n as Sum(k=0..n)T(n,k)*binomial(n,k).
28
0, 0, -1, 0, 0, 2, 0, 0, 0, -3, 0, 0, 0, -3, 16, 0, 0, 0, -3, 32, -135, 0, 0, 0, -3, 64, -405, 1536, 0, 0, 0, -3, 128, -1215, 6144, -21875, 0, 0, 0, -3, 256, -3645, 24576, -109375, 373248, 0, 0, 0, -3, 512, -10935, 98304, -546875, 2239488, -7411887, 0, 0, 0, -3, 1024, -32805, 393216, -2734375, 13436928, -51883209, 167772160
OFFSET
0,6
COMMENTS
T(n,k)=(-1)^k*k*(k-2)^(n-2) for k>1, while T(n,0)=0 and T(1,1)=-0^(n-1) 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.(21), with a=2, b=1.
EXAMPLE
The first rows of the triangle are:
0,
0, -1,
0, 0, 2,
0, 0, 0, -3,
0, 0, 0, -3, 16,
0, 0, 0, -3, 32, -135,
0, 0, 0, -3, 64, -405, 1536,
0, 0, 0, -3, 128, -1215, 6144, -21875,
PROG
(PARI) seq(nmax)={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; if(n==1, v[irow+1]=-1, v[irow+1]=0);
for(k=2, n, v[irow+k]=(-1)^k*k*(k-2)^(n-2); ); );
return(v); }
a=seq(100);
KEYWORD
sign,tabl
AUTHOR
Stanislav Sykora, Jun 23 2014
STATUS
approved