A244117 Triangle read by rows: terms of a binomial decomposition of 1 as Sum(k=0..n)T(n,k).

1, 0, 1, 0, 2, -1, 0, 3, -6, 4, 0, 4, -24, 48, -27, 0, 5, -80, 360, -540, 256, 0, 6, -240, 2160, -6480, 7680, -3125, 0, 7, -672, 11340, -60480, 134400, -131250, 46656, 0, 8, -1792, 54432, -483840, 1792000, -3150000, 2612736, -823543, 0, 9, -4608, 244944, -3483648, 20160000, -56700000, 82301184, -59295096, 16777216

Offset

0,5

Comments

T(n,k)=(1-k)^(k-1)*k^(n-k)*binomial(n,k) for k>0, while T(n,0)=0^n 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.(4), with b=1.

Example

First rows of the triangle, all summing up to 1:
1
0 1
0 2  -1
0 3  -6   4
0 4 -24  48  -27
0 5 -80 360 -540 256

Prog

(PARI) seq(nmax, b)={my(v, n, k, irow);
  v = vector((nmax+1)*(nmax+2)/2); v[1]=1;
  for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
    for(k=1, n, v[irow+k] = (1-k*b)^(k-1)*(k*b)^(n-k)*binomial(n, k); );
  ); return(v); }
  a=seq(100, 1);

Crossrefs

Cf. A244116, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Sequence in context: A077874 A286274 A230360 * A263426 A357103 A278882

Adjacent sequences: A244114 A244115 A244116 * A244118 A244119 A244120

Keyword

sign,tabl

Author

Stanislav Sykora, Jun 21 2014

Status

approved