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A274377 E.g.f. satisfies: A(x)^A(x) = exp(2*x) * A(-x)^A(-x).

%I A274377 #32 Sep 27 2016 19:04:49
%S A274377 1,1,0,1,0,16,0,736,0,67096,0,10163176,0,2306198896,0,732199108096,0,
%T A274377 309860700130816,0,168568765338224896,0,114619705107961862656,0,
%U A274377 95251358122177791486976,0,94984793274454431691503616,0,111939507886837612683516276736,0,153907136552991217284274400567296,0,244164979570216285201628515234840576,0,442692827509235885935744380253757341696,0,909667081143908558901949811564629988048896
%N A274377 E.g.f. satisfies: A(x)^A(x) = exp(2*x) * A(-x)^A(-x).
%C A274377 a(2*n+1) = 6 (mod 10) for n>1 (conjecture).
%H A274377 Paul D. Hanna, <a href="/A274377/b274377.txt">Table of n, a(n) for n = 0..401</a>
%F A274377 E.g.f.: 1 + Series_Reversion( log( sqrt( (1+x)^(1+x) / (1-x)^(1-x) ) ) ).
%F A274377 E.g.f.: 1 + Series_Reversion( (G(x) - G(-x))/2 ), where G(x) = Series_Reversion(x/LambertW(x) - 1) = (1+x)*log(1+x).
%F A274377 E.g.f.: 1 + Series_Reversion( x - Sum_{n>=1} x^(2*n+1)/(2*n*(2*n+1)) ).
%F A274377 If n is odd then a(n) ~ c * d^n * n^(n-1) / exp(n), where d = 1.37441749603820461..., c = 0.6508250221842049... . - _Vaclav Kotesovec_, Sep 22 2016
%e A274377 E.g.f.: A(x) = 1 + x + x^3/3! + 16*x^5/5! + 736*x^7/7! + 67096*x^9/9! + 10163176*x^11/11! + 2306198896*x^13/13! + 732199108096*x^15/15! + 309860700130816*x^17/17! + 168568765338224896*x^19/19! +...
%e A274377 such that A(x)^A(x) / A(-x)^A(-x) = exp(2*x).
%e A274377 RELATED SERIES.
%e A274377 A(x)^A(x) = 1 + x + 2*x^2/2! + 4*x^3/3! + 16*x^4/4! + 56*x^5/5! + 426*x^6/6! + 2262*x^7/7! + 26944*x^8/8! + 191536*x^9/9! + 3126160*x^10/10! +...+ A275764(n)*x^n/n! +...
%e A274377 Series_Reversion(A(x) - 1) = x - x^3/6 - x^5/20 - x^7/42 - x^9/72 - x^11/110 - x^13/156 - x^15/210 - x^17/272 +...+ -x^(2*n+1)/(2*n*(2*n+1)) +...
%e A274377 Also,
%e A274377 Series_Reversion(A(x) - 1) = (G(x) - G(-x))/2, where G(x) = (1+x)*log(1+x) = Series_Reversion(x/LambertW(x) - 1), and begins:
%e A274377 G(x) = x + x^2/2 - x^3/6 + x^4/12 - x^5/20 + x^6/30 - x^7/42 + x^8/56 - x^9/72 + x^10/90 - x^11/110 + x^12/132 +...+ (-x)^n/(n*(n-1)) +...
%e A274377 GENERATING METHOD.
%e A274377 Start with a(0)=1, a(1)=1, and set a(2*n)=0 for n>0, then use the following criterion to determine the odd-indexed terms.
%e A274377 Given partial sum A(x,2*n) = Sum_{k=0..2*n} a(k)*x^k/k!, and sufficiently large N, the odd-indexed term a(2*n+1) satisfies:
%e A274377 if t > a(2*n+1)/(2*n+1)!, then
%e A274377 t > [x^(2*n+1)] ( A(x,2*n) +  t*x^(2*n+1) )^(1-1/N)
%e A274377 else if t <= a(2*n+1)/(2*n+1)! , then
%e A274377 t < [x^(2*n+1)] ( A(x,2*n) + t*x^(2*n+1) )^(1-1/N);
%e A274377 this criterion defines each term of this sequence for n>1.
%e A274377 Using the same method as above, but without setting even-indexed terms to zero, generates x/LambertW(x), e.g.f. of A177885.
%e A274377 RELATED SERIES.
%e A274377 log(A(x)) = x - x^2/2! + 3*x^3/3! - 10*x^4/4! + 60*x^5/5! - 346*x^6/6! + 3108*x^7/7! - 25600*x^8/8! + 306120*x^9/9! - 3283696*x^10/10! + 49021368*x^11/11! - 648526000*x^12/12! + 11606584080*x^13/13! - 182697457216*x^14/14! +...
%o A274377 (PARI) {a(n) = my(A = 1 + serreverse(x - sum(m=1,n\2+1, x^(2*m+1)/(2*m*(2*m+1)) ) +x^2*O(x^n) ) ); n!*polcoeff(A,n)}
%o A274377 for(n=0,40,print1(a(n),", "))
%o A274377 (PARI) /* Generating method (using sufficiently large N and precision) */
%o A274377 \p100
%o A274377 {a(n) = my(N=10^(3*n), A=[1,1]); for(i=0,n\2, A=concat(A,[0,0]); A[#A] = round( (#A-1)!*polcoeff( N*1.* Ser(A)^(1-1/N), #A-1) )/(#A-1)! ); n!*A[n+1]}
%o A274377 for(n=0,40,print1(a(n),", "))
%Y A274377 Cf. A177885, A275759, A275764.
%K A274377 nonn
%O A274377 0,6
%A A274377 _Paul D. Hanna_, Aug 23 2016