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A273953 E.g.f. satisfies A(x) = Sum_{n>=0} x^n/n! * exp(n/2*x) * A(x)^(n/2).

%I A273953 #24 Feb 16 2025 08:33:36
%S A273953 1,1,3,13,77,581,5347,58213,732937,10487737,168217811,2990748509,
%T A273953 58397418037,1242643927357,28627000014355,709933328752981,
%U A273953 18859531958840273,534365880859577777,16087267158157316323,512844446937529664173,17259468942471032848861,611530055485070740134901,22755171133646348369448323,887228501593124485460914373,36173480392953890421156056665,1539307965110263598673884269801,68247672532254821767545000249907
%N A273953 E.g.f. satisfies A(x) = Sum_{n>=0} x^n/n! * exp(n/2*x) * A(x)^(n/2).
%H A273953 Seiichi Manyama, <a href="/A273953/b273953.txt">Table of n, a(n) for n = 0..411</a>
%H A273953 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A273953 E.g.f.: 4*LambertW(-x/2*exp(x/2))^2 / (x^2*exp(x)).
%F A273953 E.g.f.: exp( L(x) ) where L(x) = -2*LambertW(-x*exp(x/2)/2) is the e.g.f. of A100526.
%F A273953 a(n) ~ sqrt(1+LambertW(exp(-1)))*n^(n-1)/(2^(n-1)*exp(n-2)*LambertW(exp(-1))^n). - _Vaclav Kotesovec_, Jun 23 2016
%F A273953 From _Seiichi Manyama_, Feb 11 2023: (Start)
%F A273953 E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^(1/2) ).
%F A273953 a(n) = (1/2^(n-1)) * Sum_{k=0..n} k^(n-k) * (k+2)^(k-1) * binomial(n,k). (End)
%e A273953 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 77*x^4/4! + 581*x^5/5! + 5347*x^6/6! + 58213*x^7/7! + 732937*x^8/8! + 10487737*x^9/9! + 168217811*x^10/10! + 2990748509*x^11/11! + 58397418037*x^12/12! +...
%e A273953 such that
%e A273953 A(x) = 1 + x*exp(x/2)*A(x)^(1/2) + x^2/2!*exp(x)*A(x) + x^3/3!*exp(3*x/2)*A(x)^(3/2) + x^4/4!*exp(2*x)*A(x)^2 + x^5/5!*exp(5*x/2)*A(x)^(5/2) + x^6/6!*exp(3*x)*A(x)^3 +...
%e A273953 The logarithm of A(x) begins:
%e A273953 log(A(x)) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! + 1476*x^6/6! + 14728*x^7/7! + 173216*x^8/8! + 2346480*x^9/9! + 35981200*x^10/10! + 616111056*x^11/11! + 11652662880*x^12/12! +...+ A100526(n)*x^n/n! +...
%e A273953 which equals -2*LambertW(-x*exp(x/2)/2).
%t A273953 CoefficientList[Series[4*LambertW[-x/2*E^(x/2)]^2 / (x^2*E^x), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jun 23 2016 *)
%o A273953 (PARI) {a(n) = my(A=1+x); for(i=1,n, A = sum(m=0,n,x^m/m!*exp(m/2*x +x*O(x^n))*A^(m/2)) ); n!*polcoeff(A,n)}
%o A273953 for(n=0,30,print1(a(n),", "))
%o A273953 (PARI) a(n) = sum(k=0, n, k^(n-k)*(k+2)^(k-1)*binomial(n, k))/2^(n-1); \\ _Seiichi Manyama_, Feb 11 2023
%Y A273953 Cf. A273954, A100526.
%K A273953 nonn
%O A273953 0,3
%A A273953 _Paul D. Hanna_, Jun 14 2016