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A127548 O.g.f.: Sum_{n>=0} n!*(x/(1+x)^2)^n.

%I A127548 #38 Sep 20 2024 02:08:51
%S A127548 1,1,0,1,4,19,112,771,6088,54213,537392,5867925,69975308,904788263,
%T A127548 12607819040,188341689287,3002539594128,50878366664393,
%U A127548 913161208490016,17304836525709097,345279674107957524,7235298537356113339
%N A127548 O.g.f.: Sum_{n>=0} n!*(x/(1+x)^2)^n.
%C A127548 a(n+1) = inverse binomial transform of A013999 = Sum_{k=0..n} binomial(n,k)*(-1)^(n-k)*A013999(k). - _Emanuele Munarini_, Jul 01 2013
%H A127548 Seiichi Manyama, <a href="/A127548/b127548.txt">Table of n, a(n) for n = 0..450</a>
%F A127548 a(n) = Sum_{s=1..n} (-1)^(n-s)*s!*C(s+n-1,2s-1) if n>=1, where C(a,b)=binomial(a,b). - _R. J. Mathar_, Jul 13 2007
%F A127548 G.f.: Q(0) where Q(k) = 1 + (2*k + 1)*x/( (1+x)^2- 2*x*(1+x)^2*(k+1)/(2*x*(k+1) + (1+x)^2/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Mar 08 2013
%F A127548 a(n) = A000271(n) + A000271(n-1). - _Peter Bala_, Sep 02 2016
%F A127548 a(n) ~ exp(-2) * n!. - _Vaclav Kotesovec_, Oct 31 2017
%p A127548 A127548 := proc(n) if n = 0 then 1 ; else add(factorial(s)*(-1)^(n-s)*binomial(s+n-1,2*s-1),s=1..n) ; fi ; end: for n from 0 to 20 do printf("%d,",A127548(n)) ; od ; # _R. J. Mathar_, Jul 13 2007
%t A127548 nn = 21; CoefficientList[Series[Sum[n!*(x/(1 + x)^2)^n, {n, 0, nn}], {x, 0, nn}], x] (* _Michael De Vlieger_, Sep 04 2016 *)
%o A127548 (Python)
%o A127548 import math
%o A127548 def binomial(n,m):
%o A127548     a=1
%o A127548     for k in range(n-m+1,n+1):
%o A127548         a *= k
%o A127548     return a//math.factorial(m)
%o A127548 def A127548(n):
%o A127548     if n == 0:
%o A127548         return 1
%o A127548     a=0
%o A127548     for s in range(1,n+1):
%o A127548         a += (-1)**(n-s)*binomial(s+n-1,2*s-1)*math.factorial(s)
%o A127548     return a
%o A127548 for n in range(30):
%o A127548     print(A127548(n))
%o A127548 # _R. J. Mathar_, Oct 20 2009
%Y A127548 Cf. A000179, A078480, A013999, A000271, A000904, A078509.
%K A127548 easy,nonn
%O A127548 0,5
%A A127548 _Vladeta Jovovic_, Jun 27 2007
%E A127548 More terms from _R. J. Mathar_, Jul 13 2007
%E A127548 More terms from _R. J. Mathar_, Oct 20 2009