A244116 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 1 as Sum_{k=0..n} T(n,k)*binomial(n,k).

1, 0, 1, 0, 1, -1, 0, 1, -2, 4, 0, 1, -4, 12, -27, 0, 1, -8, 36, -108, 256, 0, 1, -16, 108, -432, 1280, -3125, 0, 1, -32, 324, -1728, 6400, -18750, 46656, 0, 1, -64, 972, -6912, 32000, -112500, 326592, -823543, 0, 1, -128, 2916, -27648, 160000, -675000, 2286144, -6588344, 16777216

Offset

0,9

Comments

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

Links

Stanislav Sykora, Table of n, rows 0..100

S. Sykora, An Abel's Identity and its Corollaries, Stan's Library, Volume V, 2014. See eq.(4) with b=1.

Example

The first few rows of the triangle are:
  1
  0 1
  0 1 -1
  0 1 -2 4
  0 1 -4 12  -27
  0 1 -8 36 -108 256
  ...

Maple

A244116 := (n, j) -> (-1)^(j + 1) * j^(n - j) * (j - 1)^(j - 1):
for n from 0 to 9 do seq(A244116(n, k), k = 0..n) od; # Peter Luschny, Jan 28 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); );
  ); return(v); }
  a=seq(100, 1);

Crossrefs

Cf. 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, A244142, A244143.

Cf. A357247.

Sequence in context: A351761 A226031 A308460 * A138133 A302937 A363367

Adjacent sequences: A244113 A244114 A244115 * A244117 A244118 A244119

Keyword

sign,tabl

Author

Stanislav Sykora, Jun 21 2014

Status

approved