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

0,5

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.

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.

Example

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

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

Sequence in context: A086468 A125507 A198583 * A280652 A104538 A120580

Adjacent sequences: A244120 A244121 A244122 * A244124 A244125 A244126

Keyword

sign,tabl

Author

Stanislav Sykora, Jun 21 2014

Status

approved