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%I
%S 1,3,1,8,9,1,42,59,18,1,144,450,215,30,1,1440,3394,2475,565,45,1,5760,
%T 30912,28294,9345,1225,63,1,75600,293292,340116,147889,27720,2338,84,
%U 1,524160,3032208,4335596,2341332,579369,69552,4074,108,1,6531840,36290736,57773700,38049920,11744775,1857513,154350,6630,135,1
%N Triangle of D'Arcais numbers.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 159.
%F G.f.: Sum_{1<=k<=n} T(n, k)*u^k*t^n/n! = ((1-t)*(1-t^2)*(1-t^3)...)^(-u).
%F Recurrence for degree n D'Arcais polynomials T(n; u) = Sum_{k=1..n} T(n, k)*u^k is given by T(n; u) = Sum_{k=1..n} (n-1)!/(n-k)!*sigma(k)*u*T(n-k; u), T(0; u) = 1. - _Vladeta Jovovic_, Oct 11 2002
%F T(n; u) = n!*Sum_{pi} Product_{i=1..n} binomial(u+k(i)-1, k(i)) where pi runs through all nonnegative solutions of k(1)+2*k(2)+..+n*k(n)=n. - _Vladeta Jovovic_, Oct 11 2002
%F E.g.f.: exp(Sum_{n>0} sigma(n)*u*x^n/n), where sigma(n)=A000203(n). - _Vladeta Jovovic_, Jan 10 2003
%e exp(Sum_{n>0} sigma(n)*u*x^n/n) = 1+u*x/1!+(3*u+u^2)*x^2/2!+(8*u+9*u^2+u^3)*x^3/3!+(42*u+59*u^2+18*u^3+u^4)*x^4/4!+...
%e 1; 3,1; 8,9,1; 42,59,18,1; ...
%e T(4; u) = 4!*(binomial(u+3,4)+binomial(u+1,2)*binomial(u,1)+binomial(u+1,2)+binomial(u,1)^2+binomial(u,1)) = 42*u+59*u^2+18*u^3+u^4.
%t t[0][u_] = 1; t[n_][u_] := t[n][u] = Sum[(n-1)!/(n-k)!*DivisorSigma[1, k]*u*t[n-k][u], {k, 1, n}]; row[n_] := CoefficientList[ t[n][u], u] // Rest; Table[row[n], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Oct 03 2012, after _Vladeta Jovovic_ *)
%Y Diagonals give A038048, A059356, A059357.
%Y Row sums give A053529.
%K nonn,tabl,nice,easy,changed
%O 1,2
%A _N. J. A. Sloane_.
%E More terms from _Vladeta Jovovic_, Dec 28 2001
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