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

%I A244129 #18 Sep 13 2017 17:28:57
%S A244129 0,1,0,2,-2,0,3,-12,9,0,4,-48,108,-64,0,5,-160,810,-1280,625,0,6,-480,
%T A244129 4860,-15360,18750,-7776,0,7,-1344,25515,-143360,328125,-326592,
%U A244129 117649,0,8,-3584,122472,-1146880,4375000,-7838208,6588344,-2097152
%N A244129 Triangle read by rows: terms of a binomial decomposition of 0^(n-1) as Sum(k=0..n)T(n,k).
%C A244129 T(n,k) = (-k)^(k-1) * k^(n-k) * binomial(n,k) for k>0, while T(n,0)=0 by convention.
%H A244129 Stanislav Sykora, <a href="/A244129/b244129.txt">Table of n, a(n) for rows 1..100</a>
%H A244129 S. Sykora, <a href="http://dx.doi.org/10.3247/SL5Math14.004">An Abel's Identity and its Corollaries</a>, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(11), with b=1.
%F A244129 E.g.f. A(x,y) satisfies: A(x,y) * exp( A(x,y) ) = y*x*exp(x). - _Paul D. Hanna_, Sep 13 2017
%e A244129 First rows of the triangle, starting at row n=1. All rows sum up to 0, except the first one whose sum is 1:
%e A244129 0, 1;
%e A244129 0, 2, -2;
%e A244129 0, 3, -12, 9;
%e A244129 0, 4, -48, 108, -64;
%e A244129 0, 5, -160, 810, -1280, 625;
%e A244129 0, 6, -480, 4860, -15360, 18750, -7776;
%e A244129 0, 7, -1344, 25515, -143360, 328125, -326592, 117649;
%e A244129 0, 8, -3584, 122472, -1146880, 4375000, -7838208, 6588344, -2097152; ...
%e A244129 From _Paul D. Hanna_, Sep 13 2017: (Start)
%e A244129 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! +...
%e A244129 such that A(x,y) * exp( A(x,y) ) = y*x*exp(x). (End)
%o A244129 (PARI) seq(nmax, b)={my(v, n, k, irow);
%o A244129 v = vector((nmax+1)*(nmax+2)/2-1);
%o A244129 for(n=1, nmax, irow=n*(n+1)/2; v[irow]=0;
%o A244129   for(k=1, n, v[irow+k]=(-1)^(k-1)*(k*b)^(n-1)*binomial(n,k); ); );
%o A244129 return(v); }
%o A244129 a=seq(100, 1);
%Y A244129 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.
%K A244129 sign,tabl
%O A244129 1,4
%A A244129 _Stanislav Sykora_, Jun 22 2014