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

Author

Stanislav Sykora, Jun 22 2014

Keywords

• sign
• tabl

Comments

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

Examples

			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)
		

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.

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.

Programs

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);

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