A244126 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 2^n-1 as Sum(k=0..n)T(n,k)*binomial(n,k).

0, 0, 1, 0, 0, 3, 0, 0, -3, 16, 0, 0, 3, -32, 125, 0, 0, -3, 64, -375, 1296, 0, 0, 3, -128, 1125, -5184, 16807, 0, 0, -3, 256, -3375, 20736, -84035, 262144, 0, 0, 3, -512, 10125, -82944, 420175, -1572864, 4782969, 0, 0, -3, 1024, -30375, 331776, -2100875, 9437184, -33480783, 100000000, 0, 0, 3, -2048, 91125, -1327104, 10504375, -56623104, 234365481, -800000000, 2357947691, 0, 0, -3, 4096, -273375, 5308416, -52521875

Offset

0,6

Comments

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

Example

The first rows of the triangle are:
0,
0, 1,
0, 0, 3,
0, 0, -3, 16,
0, 0, 3, -32, 125,
0, 0, -3, 64, -375, 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); ); );
  return(v); }
  a=seq(100, -1)

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Sequence in context: A079201 A279368 A021773 * A378039 A133109 A130208

Adjacent sequences: A244123 A244124 A244125 * A244127 A244128 A244129

Keyword

sign,tabl

Author

Stanislav Sykora, Jun 21 2014

Status

approved