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A244123
Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).
28
1, 0, 1, 0, -4, 8, 0, 9, -90, 108, 0, -16, 576, -2352, 2048, 0, 25, -2800, 28800, -72900, 50000, 0, -36, 11520, -262440, 1440000, -2635380, 1492992, 0, 49, -42336, 1984500, -20870080, 76204800, -109160142, 52706752, 0, -64, 143360, -13172544, 247726080, -1599416000, 4337012736, -5103000000, 2147483648
OFFSET
0,5
COMMENTS
T(n,k)=n*(n+k)^(k-1)*(-k)^(n-k)*binomial(n,k) for k>0, while T(n,0)=0^n 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.(5), with b=-1.
EXAMPLE
First rows of the triangle, all summing up to n^n:
1
0 1
0 -4 8
0, 9 -90 108
0 -16 576 -2352 2048
0, 25 -2800 28800 -72900 50000
PROG
(PARI) seq(nmax, b)={my(v, n, k, irow);
v = vector((nmax+1)*(nmax+2)/2); v[1]=1;
for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
for(k=1, n, v[irow+k]=n*(n-k*b)^(k-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