A244125 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, 4, -1, 0, 12, -9, 4, 0, 32, -54, 64, -27, 0, 80, -270, 640, -675, 256, 0, 192, -1215, 5120, -10125, 9216, -3125, 0, 448, -5103, 35840, -118125, 193536, -153125, 46656, 0, 1024, -20412, 229376, -1181250, 3096576, -4287500, 2985984, -823543

Offset

0,5

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:
1
0 1
0 4  -1
0 12 -9 4
0 32 -54 64 -27
0 80 -270 640 -675 256

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, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Sequence in context: A145880 A048516 A060638 * A007789 A345393 A081114

Adjacent sequences: A244122 A244123 A244124 * A244126 A244127 A244128

Keyword

sign,tabl

Author

Stanislav Sykora, Jun 21 2014

Status

approved