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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). 28
 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 (list; table; graph; refs; listen; history; text; internal format)
 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: A336978 A011137 A143396 * A342987 A090657 A167001 Adjacent sequences:  A244126 A244127 A244128 * A244130 A244131 A244132 KEYWORD sign,tabl AUTHOR Stanislav Sykora, Jun 22 2014 STATUS approved

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Last modified April 11 16:13 EDT 2021. Contains 342886 sequences. (Running on oeis4.)