A244119 Triangle read by rows: terms of a binomial decomposition of 1 as Sum(k=0..n)T(n,k).

1, 0, 1, 0, -2, 3, 0, 3, -18, 16, 0, -4, 72, -192, 125, 0, 5, -240, 1440, -2500, 1296, 0, -6, 720, -8640, 30000, -38880, 16807, 0, 7, -2016, 45360, -280000, 680400, -705894, 262144, 0, -8, 5376, -217728, 2240000, -9072000, 16941456, -14680064, 4782969

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.

Sequence A161628, arising from a different context, appears to be the same, but with opposite signs of odd rows.

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    3
0  3  -18   16
0 -4   72 -192   125
0  5 -240 1440 -2500 1296

Maple

A244119 := (n, k) -> (1+k)^(k-1)*(-k)^(n-k)*binomial(n, k):
seq(seq(A244119(n, k), k = 0..n), n = 0..8); # Peter Luschny, Jan 29 2023

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. A161628, A244116, A244117, A244118, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Cf. A273954.

Sequence in context: A370983 A257740 A161628 * A122059 A324906 A164917

Adjacent sequences: A244116 A244117 A244118 * A244120 A244121 A244122

Keyword

sign,tabl

Author

Stanislav Sykora, Jun 21 2014

Status

approved