A244140 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n*(-1)^n as Sum(k=0..n)T(n,k)*binomial(n,k).

0, 0, -1, 0, 0, 2, 0, 0, 0, -3, 0, 0, 0, -3, 16, 0, 0, 0, -3, 32, -135, 0, 0, 0, -3, 64, -405, 1536, 0, 0, 0, -3, 128, -1215, 6144, -21875, 0, 0, 0, -3, 256, -3645, 24576, -109375, 373248, 0, 0, 0, -3, 512, -10935, 98304, -546875, 2239488, -7411887, 0, 0, 0, -3, 1024, -32805, 393216, -2734375, 13436928, -51883209, 167772160

Offset: 0

Author

Stanislav Sykora, Jun 23 2014

Keywords

• sign
• tabl

Comments

T(n,k)=(-1)^k*k*(k-2)^(n-2) for k>1, while T(n,0)=0 and T(1,1)=-0^(n-1) by convention.

Examples

			The first rows of the triangle are:
0,
0, -1,
0, 0, 2,
0, 0, 0, -3,
0, 0, 0, -3, 16,
0, 0, 0, -3, 32, -135,
0, 0, 0, -3, 64, -405, 1536,
0, 0, 0, -3, 128, -1215, 6144, -21875,
		

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.(21), with a=2, b=1.

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244141, A244142, A244143.

Programs

PARI

seq(nmax)={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;if(n==1,v[irow+1]=-1,v[irow+1]=0);
for(k=2,n,v[irow+k]=(-1)^k*k*(k-2)^(n-2);););
return(v);}
a=seq(100);