A244118 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 1 as Sum_{k=0..n} T(n,k)*binomial(n,k).

1, 0, 1, 0, -1, 3, 0, 1, -6, 16, 0, -1, 12, -48, 125, 0, 1, -24, 144, -500, 1296, 0, -1, 48, -432, 2000, -6480, 16807, 0, 1, -96, 1296, -8000, 32400, -100842, 262144, 0, -1, 192, -3888, 32000, -162000, 605052, -1835008, 4782969, 0, 1, -384, 11664, -128000, 810000, -3630312, 12845056, -38263752, 100000000

Offset: 0

Author

Stanislav Sykora, Jun 21 2014

Keywords

• sign
• tabl

Comments

T(n,k) = (1+k)^(k-1)*(-k)^(n-k) for k>0, where T(n,0) = 0^n.

Examples

			The first rows of the triangle are:
1
0  1
0 -1   3
0  1  -6  16
0 -1  12 -48  125
0  1 -24 144 -500 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.(4), with b=-1.

Crossrefs

Cf. A244116, A244117, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, 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]=1;
  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)*(k*b)^(n-k););
  );return(v);}
  a=seq(100,-1);