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%I #39 Feb 23 2023 18:02:53
%S 1,1,3,10,49,271,1861,14218,125945,1222525,13274551,155993586,
%T 2006410033,27654327715,411146102641,6498852124066,109654601938801,
%U 1954603251773273,36899871955766155,732088187170903162,15288727416236948241,334089096072180624631
%N E.g.f.: A(x) = 1/[(1 - x^2)^(1 + 1/x)].
%C More generally, we have the identity:
%C Sum_{n>=0} (x^n/n!)*Product_{k=1..n} (1+k*y) = 1/(1 - x*y)^(1 + 1/y); here y=x.
%H Alois P. Heinz, <a href="/A193281/b193281.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f.: A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} (1 + k*x).
%F E.g.f.: A(x)=1/[ (1 - x^2)*(1 - x^2)^(1/x)] = 1 + x*(1+x)/(G(0) - x*(1+x)) ; G(k) = 1 + k + x*(k*x+x+1) - x*(k+1)*(k*x+2*x+1)/G(k+1) ; (continued fraction). - _Sergei N. Gladkovskii_, Dec 02 2011
%F a(n) ~ n! * (n + log(n) + 2 + gamma - log(2))/4, where gamma is the Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Oct 08 2013
%F From _Peter Bala_, Aug 31 2014: (Start)
%F E.g.f.: exp( sum{n = 1..inf} x^n/A110654(n) ).
%F It appears that a(n) = 1 ( mod n*(n - 1) ) for n even, while a(n) = 1 ( mod n*(n - 1)/2 ) for n odd [checked up to a(100)]. (End)
%F a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k/A110654(k) * a(n-k)/(n-k)!. - _Seiichi Manyama_, Apr 30 2022
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 49*x^4/4! + 271*x^5/5! +...
%e where:
%e A(x)^(x/(1+x)) = 1 + x^2 + x^4 + x^6 + x^8 + x^10 + x^12 +...
%e Also,
%e A(x) = 1 + x*(1+x) + x^2*(1+x)*(1+2*x)/2! + x^3*(1+x)*(1+2*x)*(1+3*x)/3! + x^4*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)/4! +...
%e The logarithm begins:
%e log(A(x)) = x + x^2 + x^3/2 + x^4/2 + x^5/3 + x^6/3 + x^7/4 + x^8/4 +...
%t CoefficientList[Series[1/((1-x^2)^(1+1/x)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 08 2013 *)
%o (PARI) {a(n)=n!*polcoeff(1/(1 - x^2 +x^2*O(x^n))^((1+x)/x),n)}
%o (PARI) {a(n)=n!*polcoeff(sum(m=0,n,x^m/m!*prod(k=1,m,1+k*x+x*O(x^n))),n)}
%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j/((j+1)\2)*v[i-j+1]/(i-j)!)); v; \\ _Seiichi Manyama_, Apr 30 2022
%Y Cf. A110654, A193287, A193288, A193289, A193290, A246689.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 21 2011
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