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A240172 O.g.f.: Sum_{n>=0} n! * x^n * (1+x)^n.

%I A240172 #22 Sep 22 2016 07:22:15
%S A240172 1,1,3,10,44,234,1470,10656,87624,806280,8211000,91707120,1114793280,
%T A240172 14653936080,207138844080,3133376225280,50508380361600,
%U A240172 864341342363520,15650522186302080,298948657681094400,6007868689030387200,126719410500228268800,2799004485444175008000,64613640777996615782400
%N A240172 O.g.f.: Sum_{n>=0} n! * x^n * (1+x)^n.
%C A240172 Compare to the following identities, which hold for all fixed k:
%C A240172 (1) Sum_{n>=0} n!*x^n = Sum_{n>=0} x^n * (n + k*x)^n / (1 + n*x + k*x^2)^(n+1).
%C A240172 (2) Sum_{n>=0} n!*x^(2*n) = Sum_{n>=0} x^n * (k + n*x)^n / (1 + k*x + n*x^2)^(n+1).
%C A240172 Number of ordered set partitions of [n] such that for each block b the smallest integer interval containing b has at most two elements. a(3) = 10: 12|3, 3|12, 1|23, 23|1, 1|2|3, 1|3|2, 2|1|3, 2|3|1, 3|1|2, 3|2|1 (block 13 is not allowed here). - _Alois P. Heinz_, Sep 21 2016
%H A240172 Alois P. Heinz, <a href="/A240172/b240172.txt">Table of n, a(n) for n = 0..450</a>
%F A240172 O.g.f.: Sum_{n>=0} n^n * x^n * (1+x)^n / (1 + n*x + n*x^2)^(n+1).
%F A240172 a(n) = Sum_{k=0..[n/2]} binomial(n-k, k) * (n-k)!.
%F A240172 a(n) ~ exp(1) * n!. - _Vaclav Kotesovec_, Aug 02 2014
%e A240172 O.g.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 44*x^4 + 234*x^5 + 1470*x^6 +...
%e A240172 where
%e A240172 A(x) = 1 + x*(1+x) + 2!*x^2*(1+x)^2 + 3!*x^3*(1+x)^3 + 4!*x^4*(1+x)^4 +...
%e A240172 Also, we have the identity:
%e A240172 A(x) = 1 + x*(1+x)/(1+x+x^2)^2 + 2^2*x^2*(1+x)^2/(1+2*x+2*x^2)^3 + 3^3*x^3*(1+x)^3/(1+3*x+3*x^2)^4 + 4^4*x^4*(1+x)^4/(1+4*x+4*x^2)^5 +...
%p A240172 a:= proc(n) option remember; `if`(n<3, n^2-n+1,
%p A240172       (n-2)*a(n-1)+(2*n-1)*a(n-2)+(n-1)*a(n-3))
%p A240172     end:
%p A240172 seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 21 2016
%t A240172 Table[Sum[Binomial[n-k, k] * (n-k)!,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Aug 02 2014 *)
%o A240172 (PARI) {a(n)=local(A=1);A=sum(m=0, n, m^m*x^m*(1+x)^m/(1 + m*x + m*x^2 +x*O(x^n))^(m+1));polcoeff(A, n)}
%o A240172 for(n=0,30,print1(a(n),", "))
%o A240172 (PARI) {a(n)=sum(k=0, n\2, binomial(n-k,k) * (n-k)! )}
%o A240172 for(n=0,30,print1(a(n),", "))
%Y A240172 Column k=2 of A276890.
%K A240172 nonn
%O A240172 0,3
%A A240172 _Paul D. Hanna_, Aug 02 2014