A244132 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n as Sum(k=0..n)T(n,k)*binomial(n,k).

0, 0, 1, 0, 0, 2, 0, 0, -2, 9, 0, 0, 2, -18, 64, 0, 0, -2, 36, -192, 625, 0, 0, 2, -72, 576, -2500, 7776, 0, 0, -2, 144, -1728, 10000, -38880, 117649, 0, 0, 2, -288, 5184, -40000, 194400, -705894, 2097152, 0, 0, -2, 576, -15552, 160000, -972000, 4235364, -14680064, 43046721

Offset

0,6

Comments

T(n,k)=(k)^(k-1)*(1-k)^(n-k) for k>0, while T(n,0)=0 by convention.

Links

Stanislav Sykora, Table of n, a(n) for rows 0..100

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

The first rows of the triangle are:
0,
0, 1,
0, 0, 2,
0, 0, -2, 9,
0, 0, 2, -18, 64,
0, 0, -2, 36, -192, 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); ); );
return(v); }
a=seq(100, -1);

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Sequence in context: A202496 A165664 A019263 * A337615 A330607 A169776

Adjacent sequences: A244129 A244130 A244131 * A244133 A244134 A244135

Keyword

sign,tabl

Author

Stanislav Sykora, Jun 22 2014

Status

approved