A121452 - OEIS

%I #39 Sep 15 2024 03:34:33

%S 1,1,1,4,13,71,391,2836,21729,198829,1939501,21515836,254169301,

%T 3319328299,45979476635,691443303916,10979537304961,186915474027321,

%U 3345563762493049,63613875064443796,1266776073045809341

%N Exponential generating function (1-x^2)^(-1/x).

%H Vincenzo Librandi, <a href="/A121452/b121452.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f.: (1-x^2)^(-1/x) = Sum_{n>=0} a(n)*x^n/n!.

%F E.g.f.: exp( (1-x) * Sum_{n>=0} x^n * Sum_{k=1..n+1} x^k/k ). - _Paul D. Hanna_, May 03 2012

%F a(n) = n!*Sum_{m=floor((n+1)/2)..n} (-1)^(n-m)*Stirling1(m,2*m-n)/m!. - _Vladimir Kruchinin_, Mar 09 2013

%F a(n) ~ n! / 2. - _Vaclav Kotesovec_, Feb 25 2014

%F a(0) = 1; a(n) = (n-1)! * Sum_{k=1..floor((n+1)/2)} (2*k-1)/k * a(n-2*k+1)/(n-2*k+1)!. - _Seiichi Manyama_, May 01 2022

%e E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 71*x^5/5! + 391*x^6/6! + ... such that 1/A(x)^x = 1 - x^2.

%e The logarithm of the e.g.f. is given by the series:

%e log(A(x)) = (1-x)*(x + x*(x+x/2) + x^2*(x+x^2/2+x^3/3) + x^3*(x+x^2/2+x^3/3+x^4/4)  + x^4*(x+x^2/2+x^3/3+x^4/4+x^5/5) + ...)

%e log(A(x)) = x + x^3/2 + x^5/3 + x^7/4 + x^9/5 + ...

%t With[{nn=20},CoefficientList[Series[(1-x^2)^(-1/x),{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Sep 28 2013 *)

%o (PARI) {a(n)=n!*polcoeff((1-x^2 +x^2*O(x^n))^(-1/x),n)}

%o (PARI) {a(n)=n!*polcoeff(exp((1-x)*sum(m=0,n,x^m*sum(k=1,m+1,x^k/k)+x*O(x^n))),n)} /* _Paul D. Hanna_, May 03 2012 */

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x^2)/x))) \\ _Seiichi Manyama_, May 01 2022

%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+1)\2, (2*j-1)/j*v[i-2*j+2]/(i-2*j+1)!)); v; \\ _Seiichi Manyama_, May 01 2022

%o (PARI) a(n) = n!*sum(k=0, n\2, abs(stirling(n-k, n-2*k, 1))/(n-k)!); \\ _Seiichi Manyama_, May 01 2022

%Y Cf. A087761.

%K easy,nonn

%O 0,4

%A _Vladeta Jovovic_, Sep 07 2006

%E More terms from _Klaus Brockhaus_, Sep 10 2006