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%I #10 Aug 11 2021 16:55:21
%S 2,3,28,475,11556,362418,13820696,617990499,31613351140,1817581003238,
%T 115889207486856,8109201377360590,617515197617355688,
%U 50821356562425738180,4494201727819787034288,424930120163186457909923,42774475244179902914458980,4567011383256552019018866462,515499978801248295048365872680,61333960606400043841174213941210,7671985428777785517888000656381240
%N L.g.f. A(x) satisfies: [x^(n-1)] exp( n^2*x - n*A(x) ) = 0 for n>1.
%C a(2^k) is odd for k >= 1, and a(n) is even elsewhere.
%H Paul D. Hanna, <a href="/A319146/b319146.txt">Table of n, a(n) for n = 1..300</a>
%F a(n) ~ sqrt(1-c) * 2^(2*n - 3/2) * n^(n - 1/2) / (sqrt(Pi) * exp(n) * c^n * (2-c)^(n-1)), where c = -LambertW(-2*exp(-2)) = -A226775. - _Vaclav Kotesovec_, Aug 11 2021
%e L.g.f., A(x) = Sum_{n>=1} a(n)*x^n/n, begins
%e A(x) = 2*x + 3*x^2/2 + 28*x^3/3 + 475*x^4/4 + 11556*x^5/5 + 362418*x^6/6 + 13820696*x^7/7 + 617990499*x^8/8 + 31613351140*x^9/9 + 1817581003238*x^10/10 + ...
%e RELATED SERIES.
%e exp(A(x)) = 1 + 2*x + 7*x^2/2! + 82*x^3/3! + 3413*x^4/4! + 310306*x^5/5! + 47180827*x^6 + 10609392242*x^7/7! + 3284088709897*x^8/8! + 1333647722701378*x^9/9! + ... + A319144(n)*x^n/n! +...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k/k! in exp(n^2*x - n*A(x)) begins
%e n=1: [1, -1, -2, -48, -2616, -262080, -41718240, -9630270720, ...];
%e n=2: [1, 0, -6, -112, -5592, -547968, -86345120, -19809990912, ...];
%e n=3: [1, 3, 0, -222, -10728, -958824, -144971712, -32519314080, ...];
%e n=4: [1, 8, 52, 0, -18648, -1693248, -236690784, -50727983616, ...];
%e n=5: [1, 15, 210, 2420, 0, -2739720, -399251600, -80125144800, ...];
%e n=6: [1, 24, 558, 12192, 221184, 0, -616918320, -131299591680, ...];
%e n=7: [1, 35, 1204, 40278, 1272768, 33597312, 0, -196436730672, ...];
%e n=8: [1, 48, 2280, 106688, 4869552, 210771456, 7654459648, 0, ...]; ...
%e in which the n-th term in row n forms a diagonal of zeros after an initial '1'.
%e ODD TERMS.
%e The odd terms occur at positions 2^k for k >= 1, and begin:
%e a(2) = 3,
%e a(4) = 475,
%e a(8) = 617990499,
%e a(16) = 424930120163186457909923,
%e a(32) = 14084628818669848855677925942788864036579283666486658556963, ...
%o (PARI) {a(n) = my(A=[1],L=[2]); for(m=1, n+1, A=concat(A, 0); A[m] = Vec( exp(m^2*x +x*O(x^n))/Ser(A)^(m) )[m]/m ); L=Vec(Ser(A)'/Ser(A)); L[n]}
%o for(n=1, 21, print1(a(n), ", "))
%Y Cf. A319144, A317344.
%K nonn
%O 1,1
%A _Paul D. Hanna_, Sep 18 2018
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