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 A219344 O.g.f. satisfies: A(x) = Sum_{n>=0} 2*(n+2)^(n-1) * (n*x)^n * A(n*x)^n/n! * exp(-(n+2)*n*x*A(n*x)). 1
 1, 2, 14, 238, 6636, 273354, 15920706, 1292724636, 146453417488, 23281175674462, 5218509363479914, 1654434832566803018, 743482275590960686464, 474454676244907785390480, 430533281246889283353506596, 556106522019612061492965277720, 1023417606325146596408758881753232 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Compare the g.f. to the LambertW identity: 1 = Sum_{n>=0} 2*(n+2)^(n-1) * exp(-(n+2)*x) * x^n/n!. LINKS EXAMPLE O.g.f.: A(x) = 1 + 2*x + 14*x^2 + 238*x^3 + 6636*x^4 + 273354*x^5 +... where A(x) = 1 + 2*3^0*1^1*x*A(x)*exp(-3*1*x*A(x)) + 2*4^1*2^2*x^2*A(2*x)^2*exp(-4*2*x*A(2*x))/2! + 2*5^2*3^3*x^3*A(3*x)^3*exp(-5*3*x*A(3*x))/3! + 2*6^3*4^4*x^4*A(4*x)^4*exp(-6*4*x*A(4*x))/4! + 2*7^4*5^5*x^5*A(5*x)^5*exp(-7*5*x*A(5*x))/5! +... simplifies to a power series in x with integer coefficients. PROG (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, 2*(k+2)^(k-1)*k^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k+2)*k*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)} for(n=0, 25, print1(a(n), ", ")) CROSSREFS Cf. A219345, A218102, A219220, A217900. Sequence in context: A052215 A053846 A053855 * A343441 A152476 A070813 Adjacent sequences:  A219341 A219342 A219343 * A219345 A219346 A219347 KEYWORD nonn AUTHOR Paul D. Hanna, Nov 18 2012 STATUS approved

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Last modified July 26 14:45 EDT 2021. Contains 346294 sequences. (Running on oeis4.)