A244131 Triangle read by rows: terms T(n,k) of a binomial decomposition of n as Sum(k=0..n)T(n,k).

0, 0, 1, 0, 4, -2, 0, 12, -18, 9, 0, 32, -108, 144, -64, 0, 80, -540, 1440, -1600, 625, 0, 192, -2430, 11520, -24000, 22500, -7776, 0, 448, -10206, 80640, -280000, 472500, -381024, 117649, 0, 1024, -40824, 516096, -2800000, 7560000, -10668672, 7529536, -2097152

Offset

0,5

Comments

T(n,k)=(-k)^(k-1)*(1+k)^(n-k)*binomial(n,k) for k>0, while T(n,0)=0 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.(12), with b=1.

Example

First rows of the triangle, all summing up to n:
0,
0, 1,
0, 4, -2,
0, 12, -18, 9,
0, 32, -108, 144, -64,
0, 80, -540, 1440, -1600, 625,

Prog

(PARI) seq(nmax, b)={my(v, n, k, irow);
v = vector((nmax+1)*(nmax+2)/2); v[1]=0;
for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
  for(k=1, n, v[irow+k]=(-k*b)^(k-1)*(1+k*b)^(n-k)*binomial(n, k); ); );
return(v); }
a=seq(100, 1);

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Sequence in context: A091435 A330472 A118441 * A206428 A357012 A334778

Adjacent sequences: A244128 A244129 A244130 * A244132 A244133 A244134

Keyword

sign,tabl

Author

Stanislav Sykora, Jun 22 2014

Status

approved