A244142 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, 0, 2, 0, 0, 6, -15, 0, 0, 18, -75, 196, 0, 0, 54, -375, 1372, -3645, 0, 0, 162, -1875, 9604, -32805, 87846, 0, 0, 486, -9375, 67228, -295245, 966306, -2599051, 0, 0, 1458, -46875, 470596, -2657205, 10629366, -33787663, 91125000

Offset: 0

Author

Stanislav Sykora, Jun 23 2014

Keywords

• sign
• tabl

Comments

T(n,k)=(-1)^k*k*(2*k-1)^(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, 6, -15,
0, 0, 18, -75, 196,
0, 0, 54, -375, 1372, -3645
		

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=1, b=2.

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, A244140, A244141, 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*(2*k-1)^(n-2); ); );
return(v); }
a=seq(100);