A244130 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n as Sum_{k=0..n} T(n,k)*binomial(n,k).

0, 0, 1, 0, 2, -2, 0, 4, -6, 9, 0, 8, -18, 36, -64, 0, 16, -54, 144, -320, 625, 0, 32, -162, 576, -1600, 3750, -7776, 0, 64, -486, 2304, -8000, 22500, -54432, 117649, 0, 128, -1458, 9216, -40000, 135000, -381024, 941192, -2097152, 0, 256, -4374, 36864, -200000, 810000, -2667168, 7529536, -18874368, 43046721

Offset: 0

Author

Stanislav Sykora, Jun 22 2014

Keywords

• sign
• tabl

Comments

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

Examples

			The first rows of the triangle are:
0,
0, 1,
0, 2, -2,
0, 4, -6, 9,
0, 8, -18, 36, -64,
0, 16, -54, 144, -320, 625,
		

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.(12), with b=1.

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, 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]=(-k*b)^(k-1)*(1+k*b)^(n-k);););
return(v);}
a=seq(100,1);