A244123 Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).

1, 0, 1, 0, -4, 8, 0, 9, -90, 108, 0, -16, 576, -2352, 2048, 0, 25, -2800, 28800, -72900, 50000, 0, -36, 11520, -262440, 1440000, -2635380, 1492992, 0, 49, -42336, 1984500, -20870080, 76204800, -109160142, 52706752, 0, -64, 143360, -13172544, 247726080, -1599416000, 4337012736, -5103000000, 2147483648

Offset: 0

Author

Stanislav Sykora, Jun 21 2014

Keywords

• sign
• tabl

Comments

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

Examples

			First rows of the triangle, all summing up to n^n:
1
0 1
0 -4  8
0, 9 -90 108
0 -16 576 -2352 2048
0, 25 -2800 28800 -72900 50000
		

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.(5), with b=-1.

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Programs

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]=n*(n-k*b)^(k-1)*(k*b)^(n-k)*binomial(n, k); ); );
  return(v); }
  a=seq(100,-1);