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

Author

Stanislav Sykora, Jun 21 2014

Keywords

• sign
• tabl

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.

Examples

			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
		

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.

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.

Programs

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

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);