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A227096 Self-convolution of A013999.

%I A227096 #17 Feb 14 2023 06:30:36
%S A227096 1,2,5,20,104,632,4396,34680,307236,3026472,32849364,389704800,
%T A227096 5017492320,69678231552,1038078389376,16513758904320,279354776803200,
%U A227096 5007072973075200,94783054774919040,1889504358498754560,39565281716813111040,868194780280625779200
%N A227096 Self-convolution of A013999.
%H A227096 Alois P. Heinz, <a href="/A227096/b227096.txt">Table of n, a(n) for n = 0..200</a>
%F A227096 a(n) = sum(A013999(k)*A013999(n-k), k=0..n).
%F A227096 G.f.: sum(B(k)*k!*x^(k-2)*(1-x)^k, k>=2), where B(k) = sum(1/C(k,i), i=1..k-1).
%F A227096 a(n) ~ 2*n*n!/exp(1). - _Vaclav Kotesovec_, Jul 08 2013
%p A227096 a:= proc(n) option remember; `if`(n<6, [1, 2, 5, 20, 104, 632][n+1],
%p A227096       ((3*n+10)*(n+3)*a(n-1) -(n+13)*(n+2)^2*a(n-2)
%p A227096        +(n+3)*(4*n^2+19*n+2)*a(n-3) -2*(n+2)*(3*n^2+6*n-4)*a(n-4)
%p A227096        +(4*n^3+8*n^2-12*n-4)*a(n-5) -n*(n+3)*(n-2)*a(n-6))/(2*n+4))
%p A227096     end:
%p A227096 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jul 01 2013
%t A227096 f[n_] := Sum[Binomial[n-k+1, k] (-1)^k (n-k+1)!, {k, 0, Quotient[n+1, 2]}];
%t A227096 a[n_] := Sum[f[k] f[n-k], {k, 0, n}];
%t A227096 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 14 2023 *)
%o A227096 (Maxima) f(n):=sum(binomial(n-k+1, k)*(-1)^k*(n-k+1)!, k, 0, floor((n+1)/2)); a(n):=sum(f(k)*f(n-k), k, 0, n); makelist(a(n), n, 0, 20);
%Y A227096 Cf. A013999.
%K A227096 nonn
%O A227096 0,2
%A A227096 _Emanuele Munarini_, Jul 01 2013