A244137 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, 2, 2, 0, 12, 6, 9, 0, 108, 48, 36, 64, 0, 1280, 540, 360, 320, 625, 0, 18750, 7680, 4860, 3840, 3750, 7776, 0, 326592, 131250, 80640, 60480, 52500, 54432, 117649, 0, 6588344, 2612736, 1575000, 1146880, 945000, 870912, 941192, 2097152

Offset

0,5

Comments

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

There are many binomial decompositions of n^n, some with all terms positive like this one (see A243203). However, for every n, the terms corresponding to k=1..n in this one are exceptionally similar in value (at least on log scale).

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.(13), with b=-1.

Example

First rows of the triangle, all summing up to n^n:
1,
0, 1,
0, 2, 2,
0, 12, 6, 9,
0, 108, 48, 36, 64,
0, 1280, 540, 360, 320, 625,

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

Crossrefs

Cf. A243203, A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244138, A244139, A244140, A244141, A244142, A244143.

Sequence in context: A285783 A364240 A117270 * A354127 A181389 A369072

Adjacent sequences: A244134 A244135 A244136 * A244138 A244139 A244140

Keyword

nonn,tabl

Author

Stanislav Sykora, Jun 22 2014

Status

approved