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%I #37 Sep 15 2021 07:03:16
%S 1,3,-1,0,4,-28,188,-1368,11016,-98208,964512,-10370880,121337280,
%T -1535880960,20924455680,-305396421120,4755302899200,-78700195123200,
%U 1379748896870400,-25546854999859200,498194992408780800,-10207190048993280000,219216795045212160000
%N A diagonal of A008296.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, b(n,2).
%H Robert Israel, <a href="/A045406/b045406.txt">Table of n, a(n) for n = 2..414</a>
%F a(n) = A008296(n,2).
%F E.g.f.: ((1+x)*log(1+x))^2/2. - _Vladeta Jovovic_, Feb 20 2003
%F a(n) = sum(i=1, n-1, i^2*Stirling1(n-1, i)). - _Benoit Cloitre_, Oct 23 2004
%F If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n) = f(n,2,-2), for n>=2. - _Milan Janjic_, Dec 21 2008
%F a(n) = (-1)^(n)*(2*H(n-3)-3)*(n-3)! for n >= 3, where H(n) = Sum(j=1..n, 1/j) is the n-th harmonic number. - _Gary Detlefs_, Feb 13 2010
%F a(n) = 2*A081048(n-3)-3*(-1)^(n)*(n-3)! for n >= 3. - _Robert Israel_, Jun 28 2015
%F Sum_{k=1..n} a(k+1) * Stirling2(n,k) = n^2. - _Vaclav Kotesovec_, Sep 03 2018
%F Conjecture: D-finite with recurrence a(n) +(2*n-7)*a(n-1) +(n-4)^2*a(n-2)=0. - _R. J. Mathar_, Sep 15 2021
%p with(combinat): for n from 2 to 40 do for k from 2 to 2 do printf(`%d,`,sum(binomial(l,k)*k^(l-k)*stirling1(n,l), l=k..n)) od: od:
%p # Alternative:
%p A081048:= gfun:-rectoproc({a(0)=0,a(1)=1,a(n)=(1-2*n)*a(n-1) -(n-1)^2*a(n-2)},a(n),remember):
%p 1, seq(2*A081048(n-3)-3*(-1)^(n)*(n-3)!,n=3..50); # _Robert Israel_, Jun 29 2015
%t With[{nn=30},CoefficientList[Series[((1+x)Log[1+x])^2/2,{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Jun 04 2019 *)
%Y Cf. A081048.
%K sign,easy
%O 2,2
%A _N. J. A. Sloane_, Jan 26 2001
%E More terms from _James A. Sellers_, Jan 26 2001
%E Gary Detlefs comment changed to a formula by _Robert Israel_, Jun 28 2015
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