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

Author

Stanislav Sykora, Jun 21 2014

Keywords

• sign
• tabl

Comments

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

Examples

			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
  ...
		

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.

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.

Programs

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

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);