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

0,5

Comments

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

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.

Example

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,

Prog

(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);

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.

Sequence in context: A100240 A072690 A199454 * A180813 A194656 A283240

Adjacent sequences: A244127 A244128 A244129 * A244131 A244132 A244133

Keyword

sign,tabl

Author

Stanislav Sykora, Jun 22 2014

Status

approved