A244141 Triangle read by rows: terms T(n,k) of a binomial decomposition of n*(-1)^n as Sum(k=0..n)T(n,k).

0, 0, -1, 0, 0, 2, 0, 0, 0, -3, 0, 0, 0, -12, 16, 0, 0, 0, -30, 160, -135, 0, 0, 0, -60, 960, -2430, 1536, 0, 0, 0, -105, 4480, -25515, 43008, -21875, 0, 0, 0, -168, 17920, -204120, 688128, -875000, 373248, 0, 0, 0, -252, 64512, -1377810, 8257536, -19687500, 20155392, -7411887

Offset

0,6

Comments

T(n,k)=(-1)^k*k*(k-2)^(n-2)*binomial(n,k) for k>1, while T(n,0)=0 and T(1,1)=-0^(n-1) 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.(21), with a=2, b=1.

Example

First rows of the triangle, all summing up to n*(-1)^n:
0,
0, -1,
0, 0, 2,
0, 0, 0, -3,
0, 0, 0, -12, 16,
0, 0, 0, -30, 160, -135,
0, 0, 0, -60, 960, -2430, 1536,

Prog

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

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244142, A244143.

Sequence in context: A035444 A376417 A376418 * A152489 A143655 A173541

Adjacent sequences: A244138 A244139 A244140 * A244142 A244143 A244144

Keyword

sign,tabl

Author

Stanislav Sykora, Jun 23 2014

Status

approved