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A059373 Second diagonal of triangle in A059370.

%I A059373 #12 Jun 14 2013 05:21:32
%S A059373 1,-4,8,-16,12,-96,-480,-4672,-45520,-493120,-5798912,-73668608,
%T A059373 -1005335552,-14671085568,-228051746304,-3762955404288,
%U A059373 -65707303602432,-1210821292674048,-23487031074109440,-478463919131627520,-10214440549929047040,-228069193578011566080
%N A059373 Second diagonal of triangle in A059370.
%D A059373 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 171, #34.
%F A059373 G.f. A(x) is (R(x))^2 where R(x) is the series reversion of x*hypergeom([1,2],[],x) = sum(n>=1, n!*x^n), see Comtet. - _Mark van Hoeij_, Apr 20 2013
%p A059373 series(RootOf(T*hypergeom([1,2],[],T)-x,T)^2,x=0,21); # _Mark van Hoeij_, Apr 20 2013
%t A059373 nmax = 23; t[n_, k_] := t[n, k] = Sum[(m+1)!*t[n-m-1, k-1], {m, 0, n-k}]; t[n_, 1] = n!; t[n_, n_] = 1; tnk = Table[t[n, k], {n, 1, nmax}, {k, 1, nmax}]; A059370 = Reverse /@ Inverse[tnk] // DeleteCases[#, 0, 2] & ; Table[A059370[[n, n - 1]], {n, 2, nmax}] (* _Jean-François Alcover_, Jun 14 2013 *)
%o A059373 (PARI)
%o A059373 N = 66;  x = 'x + O('x^N);
%o A059373 tf = sum(n=1,N, n!*x^n );
%o A059373 gf=serreverse(%)^2;
%o A059373 v = Vec(gf)
%o A059373 /* _Joerg Arndt_, Apr 20 2013 */
%Y A059373 Cf. A059370.
%K A059373 sign,easy
%O A059373 2,2
%A A059373 _N. J. A. Sloane_, Jan 28 2001
%E A059373 Added more terms, _Mark van Hoeij_ and _Joerg Arndt_, Apr 20 2013