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A274482 E.g.f. satisfies: A(x)^2 = A( x^2*exp(2*x) ). 1

%I

%S 1,2,6,40,320,2976,35392,538112,9810432,200903680,4480990976,

%T 107974474752,2804574622720,78474060105728,2366867031521280,

%U 77011634301730816,2702517873238802432,102101344967992541184,4139003906606265204736,179219001275949107118080,8245453502533567172837376,400905721912603103762317312,20495451270608555635778256896,1096768710243163144517540904960,61205572460929693462200057856000,3551250424196976573288927331352576,213729944352034835154759793458020352,13318442043189965481553192153652396032

%N E.g.f. satisfies: A(x)^2 = A( x^2*exp(2*x) ).

%C Compare g.f. to: C(x)^2 = C( x^2/(1 - 2*x)^2 ) when C(x) = (1-2*x - sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers (A000108).

%H Vaclav Kotesovec, <a href="/A274482/b274482.txt">Table of n, a(n) for n = 1..250</a>

%F E.g.f. A(x) satisfies: A( LambertW(x) )^2 = A(x^2).

%e E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 40*x^4/4! + 320*x^5/5! + 2976*x^6/6! + 35392*x^7/7! + 538112*x^8/8! + 9810432*x^9/9! + 200903680*x^10/10! + 4480990976*x^11/11! + 107974474752*x^12/12! +...

%e such that A( x^2*exp(2*x) ) = A(x)^2.

%e RELATED SERIES.

%e A(x)^2 = 2*x^2/2! + 12*x^3/3! + 72*x^4/4! + 640*x^5/5! + 6960*x^6/6! + 85344*x^7/7! + 1226624*x^8/8! + 21007872*x^9/9! + 419834880*x^10/10! + 9457930240*x^11/11! + 233914103808*x^12/12! +...

%e A( LambertW(x) ) = x + 3*x^3/3! + 45*x^5/5! + 3255*x^7/7! + 341145*x^9/9! + 54064395*x^11/11! + 14152823685*x^13/13! + 5781948947775*x^15/15! + 3250182676165425*x^17/17! + 2276432340328221075*x^19/19! + 1912646738585104847325*x^21/21! + 1908799239468337485243975*x^23/23! +...

%e which equals sqrt( A(x^2) ).

%o (PARI) {a(n) = my(A=x); for(i=1, #binary(n+1), A = sqrt( subst(A, x, x^2*exp(2*x +x*O(x^n)) ) ) ); n!*polcoeff(A, n)}

%o for(n=1,40,print1(a(n),", "))

%K nonn

%O 1,2

%A _Paul D. Hanna_, Aug 11 2016

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Last modified October 20 00:34 EDT 2021. Contains 348097 sequences. (Running on oeis4.)