A244129 Triangle read by rows: terms of a binomial decomposition of 0^(n-1) as Sum(k=0..n)T(n,k).

0, 1, 0, 2, -2, 0, 3, -12, 9, 0, 4, -48, 108, -64, 0, 5, -160, 810, -1280, 625, 0, 6, -480, 4860, -15360, 18750, -7776, 0, 7, -1344, 25515, -143360, 328125, -326592, 117649, 0, 8, -3584, 122472, -1146880, 4375000, -7838208, 6588344, -2097152

Offset

1,4

Comments

T(n,k) = (-k)^(k-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 1..100

S. Sykora, An Abel's Identity and its Corollaries, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(11), with b=1.

Formula

E.g.f. A(x,y) satisfies: A(x,y) * exp( A(x,y) ) = y*x*exp(x). - Paul D. Hanna, Sep 13 2017

Example

First rows of the triangle, starting at row n=1. All rows sum up to 0, except the first one whose sum is 1:
0, 1;
0, 2, -2;
0, 3, -12, 9;
0, 4, -48, 108, -64;
0, 5, -160, 810, -1280, 625;
0, 6, -480, 4860, -15360, 18750, -7776;
0, 7, -1344, 25515, -143360, 328125, -326592, 117649;
0, 8, -3584, 122472, -1146880, 4375000, -7838208, 6588344, -2097152; ...
From Paul D. Hanna, Sep 13 2017: (Start)
E.g.f.: A(x,y) = y*x + (-2*y^2 + 2*y)*x^2/2! + (9*y^3 - 12*y^2 + 3*y)*x^3/3! + (-64*y^4 + 108*y^3 - 48*y^2 + 4*y)*x^4/4! + (625*y^5 - 1280*y^4 + 810*y^3 - 160*y^2 + 5*y)*x^5/5! + (-7776*y^6 + 18750*y^5 - 15360*y^4 + 4860*y^3 - 480*y^2 + 6*y)*x^6/6! + (117649*y^7 - 326592*y^6 + 328125*y^5 -  143360*y^4 + 25515*y^3 - 1344*y^2 + 7*y)*x^7/7! +...
such that A(x,y) * exp( A(x,y) ) = y*x*exp(x). (End)

Prog

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

Crossrefs

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

Sequence in context: A143396 A350266 A361893 * A363907 A342987 A090657

Adjacent sequences: A244126 A244127 A244128 * A244130 A244131 A244132

Keyword

sign,tabl

Author

Stanislav Sykora, Jun 22 2014

Status

approved