A244124 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, 2, -1, 0, 4, -3, 4, 0, 8, -9, 16, -27, 0, 16, -27, 64, -135, 256, 0, 32, -81, 256, -675, 1536, -3125, 0, 64, -243, 1024, -3375, 9216, -21875, 46656, 0, 128, -729, 4096, -16875, 55296, -153125, 373248, -823543, 0, 256, -2187, 16384, -84375, 331776, -1071875, 2985984, -7411887, 16777216

Offset

0,5

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 2  -1
0 4  -3  4
0 8  -9  16 -27
0 16 -27 64 -135 256

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); ); );
  return(v); }
  a=seq(100, 1)

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Sequence in context: A361989 A218599 A051623 * A183190 A296129 A276544

Adjacent sequences: A244121 A244122 A244123 * A244125 A244126 A244127

Keyword

sign,tabl

Author

Stanislav Sykora, Jun 21 2014

Status

approved