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A007121 Expansion of e.g.f. ( (1+x)^x )^x.

%I A007121 M4099 #38 Jul 09 2022 11:04:48
%S A007121 1,0,0,6,-12,40,180,-1512,11760,-38880,20160,2106720,-22381920,
%T A007121 173197440,-703999296,-1737489600,86030380800,-1149696737280,
%U A007121 11455162974720,-89560399541760,636617260339200,-6318191386644480,139398889956480000,-3797936822885990400
%N A007121 Expansion of e.g.f. ( (1+x)^x )^x.
%D A007121 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007121 Seiichi Manyama, <a href="/A007121/b007121.txt">Table of n, a(n) for n = 0..452</a>
%F A007121 a(n) = n!*Sum_{k=0..floor(n/3)} Stirling1(n-2*k,k)/(n-2*k)!. - _Vladimir Kruchinin_, Dec 13 2011
%F A007121 a(0) = 1; a(n) = -(n-1)! * Sum_{k=3..n} (-1)^k * k/(k-2) * a(n-k)/(n-k)!. - _Seiichi Manyama_, Jul 09 2022
%p A007121 A007121 := proc(n)
%p A007121         n!*coeftayl( (1+x)^(x^2),x=0,n) ;
%p A007121 end proc:
%p A007121 seq(A007121(n),n=0..40) ; # _R. J. Mathar_, Dec 15 2011
%t A007121 With[{nn=30},CoefficientList[Series[((1+x)^x)^x,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Aug 24 2014 *)
%o A007121 (Maxima)
%o A007121 a(n):=sum(stirling1(n-2*k, k)/(n-2*k)!, k, 0, n/3); /* Vladimir Kruchinin, Dec 13 2011 */
%o A007121 (PARI) a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 1)/(n-2*k)!); \\ _Seiichi Manyama_, Jul 09 2022
%o A007121 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!*sum(j=3, i, (-1)^j*j/(j-2)*v[i-j+1]/(i-j)!)); v; \\ _Seiichi Manyama_, Jul 09 2022
%Y A007121 Cf. A240989.
%K A007121 sign
%O A007121 0,4
%A A007121 _Simon Plouffe_
%E A007121 Signs added by _R. J. Mathar_, _Vladimir Kruchinin_, Dec 15 2011