A244127 Triangle read by rows: terms T(n,k) of a binomial decomposition of 2^n-1 as Sum(k=0..n)T(n,k).

0, 0, 1, 0, 0, 3, 0, 0, -9, 16, 0, 0, 18, -128, 125, 0, 0, -30, 640, -1875, 1296, 0, 0, 45, -2560, 16875, -31104, 16807, 0, 0, -63, 8960, -118125, 435456, -588245, 262144, 0, 0, 84, -28672, 708750, -4644864, 11764900, -12582912, 4782969

Offset

0,6

Comments

T(n,k)=(1+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.(6), with b=-1 and a=1.

Example

First rows of the triangle, all summing up to 2^n-1:
0,
0, 1,
0, 0, 3,
0, 0, -9, 16,
0, 0, 18, -128, 125,
0, 0, -30, 640, -1875, 1296,

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)*binomial(n, k); ); );
  return(v); }
  a=seq(100, -1)

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Sequence in context: A265828 A177016 A346811 * A363407 A342312 A123474

Adjacent sequences: A244124 A244125 A244126 * A244128 A244129 A244130

Keyword

sign,tabl

Author

Stanislav Sykora, Jun 21 2014

Status

approved