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 A122525 Triangle read by rows: G(s,rho) = ((s-1)!/s)*Sum(((s-i)/i!)*(s*rho)^i, i=0..(s-1)). 5
 1, 1, 1, 2, 4, 3, 6, 18, 24, 16, 24, 96, 180, 200, 125, 120, 600, 1440, 2160, 2160, 1296, 720, 4320, 12600, 23520, 30870, 28812, 16807, 5040, 35280, 120960, 268800, 430080, 516096, 458752, 262144, 40320, 322560, 1270080, 3265920 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS When s is a positive integer and 00 and rho>0 one can use the expression: G(s,rho)=(exp(s*rho)*s*rho*(1-rho)*(s-1)*GAMMA(s-1,s*rho)+rho^s*s^s)/s/rho; For s>0 and rho>0 one can also use the integral representation G(s,rho)=(s*rho)^s/s*Int(rho*s*exp(-rho*s*t)*t*(1+t)^(s-1), t=0..infinity); Multiplying the n-th row entries by n+1 results in triangle A066324 in row reversed form. - Peter Bala, Sep 30 2011 Row generating polynomials are given by 1/n*D^n(1/(1-x*t)) evaluated at x = 0, where D is the operator exp(x)/(1-x)*d/dx. - Peter Bala, Dec 27 2011 EXAMPLE G(5,rho) = 24+96*rho+180*rho^2+200*rho^3+125*rho^4. The coefficients (24, 96, 180, 200, 125) give the 5th line of the triangle. Triangle begins: 1; 1, 1; 2, 4, 3; 6, 18, 24, 16; 24, 96, 180, 200, 125; 120, 600, 1440, 2160, 2160, 1296; 720, 4320, 12600, 23520, 30870, 28812, 16807; 5040, 35280, 120960, 268800, 430080, 516096, 458752, 262144; 40320, 322560, 1270080, 3265920, 6123600, 8817984, 9920232, 8503056, 4782969; MAPLE G:= proc(s) G(s):= (s-1)!/s*add((s-i)/i!*(s*rho)^i, i=0..(s-1)) end: T:= n-> coeff(G(n), rho, k): seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Sep 08 2012 MATHEMATICA nn=6; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; f[list_]:=Select[list, #>0&]; Map[f, Map[Reverse, Range[0, nn]!CoefficientList[Series[Exp[t]/(1-y t), {x, 0, nn}], {x, y}]]]//Grid  (* Geoffrey Critzer, Sep 08 2012 *) CROSSREFS Cf. A066324. Sequence in context: A329902 A091274 A330745 * A054589 A051851 A336165 Adjacent sequences:  A122522 A122523 A122524 * A122526 A122527 A122528 KEYWORD nonn,tabl AUTHOR Arie Harel, Sep 14 2006 STATUS approved

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Last modified April 22 19:11 EDT 2021. Contains 343177 sequences. (Running on oeis4.)