A244128 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 0^(n-1) as Sum(k=0..n)T(n,k)*binomial(n,k).

0, 1, 0, 1, -2, 0, 1, -4, 9, 0, 1, -8, 27, -64, 0, 1, -16, 81, -256, 625, 0, 1, -32, 243, -1024, 3125, -7776, 0, 1, -64, 729, -4096, 15625, -46656, 117649, 0, 1, -128, 2187, -16384, 78125, -279936, 823543, -2097152, 0, 1, -256, 6561, -65536, 390625, -1679616, 5764801, -16777216, 43046721

Offset: 1

Author

Stanislav Sykora, Jun 22 2014

Keywords

• sign
• tabf

Comments

T(n,k)=(-k)^(k-1)*k^(n-k) for k>0, while T(n,0)=0 by convention. The flattened triangle start with row 1, coefficient T(1,0).

Resembles A076014, but with added powers of 0, and with sign-alternating columns.

Examples

			The first rows of the triangle (starting at n=1):
0, 1,
0, 1, -2,
0, 1, -4, 9,
0, 1, -8, 27, -64,
0, 1, -16, 81, -256, 625,
0, 1, -32, 243, -1024, 3125, -7776,
		

Links

• Stanislav Sykora, Table of n, a(n) for rows 1..100
• S. Sykora, An Abel's Identity and its Corollaries, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(11), with b=1.

Crossrefs

Cf. A076014, A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, 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)/2-1);
for(n=1,nmax,irow=n*(n+1)/2;v[irow]=0;
  for(k=1,n,v[irow+k]=(-1)^(k-1)*(k*b)^(n-1);););
return(v);}
a=seq(100,1);