A244133 Triangle read by rows: terms T(n,k) of a binomial decomposition of n as Sum(k=0..n)T(n,k).

0, 0, 1, 0, 0, 2, 0, 0, -6, 9, 0, 0, 12, -72, 64, 0, 0, -20, 360, -960, 625, 0, 0, 30, -1440, 8640, -15000, 7776, 0, 0, -42, 5040, -60480, 210000, -272160, 117649, 0, 0, 56, -16128, 362880, -2240000, 5443200, -5647152, 2097152, 0, 0, -72, 48384, -1959552, 20160000, -81648000, 152473104, -132120576, 43046721

Offset

0,6

Comments

T(n,k)=(k)^(k-1)*(1-k)^(n-k)*binomial(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

First rows of the triangle, all summing up to n:
0,
0, 1,
0, 0, 2,
0, 0, -6, 9,
0, 0, 12, -72, 64,
0, 0, -20, 360, -960, 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)*binomial(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, A244132, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Sequence in context: A230250 A137250 A329290 * A348639 A244142 A161800

Adjacent sequences: A244130 A244131 A244132 * A244134 A244135 A244136

Keyword

sign,tabl

Author

Stanislav Sykora, Jun 22 2014

Status

approved