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

Author

Stanislav Sykora, Jun 21 2014

Keywords

• sign
• tabl

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.

Examples

			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,
		

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.

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.

Programs

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)