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%I #12 Aug 11 2021 16:48:27
%S 1,2,7,82,3413,310306,47180827,10609392242,3284088709897,
%T 1333647722701378,686179134994911311,435599748089861536402,
%U 334122749226062422725277,304457064400271021354494562,324970210527067394401358110243,401523372735670670696974799321266,568490192646838149936392483264664977,914248292513326978923735948784457567362
%N E.g.f. A(x) satisfies: [x^(n-1)] exp(n^2*x) / A(x)^n = 0 for n>1.
%C It is remarkable that the logarithmic derivative of the e.g.f. A(x) should be an integer series.
%H Paul D. Hanna, <a href="/A319144/b319144.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) ~ sqrt(1-c) * 2^(2*n - 1) * n^(2*n - 1) / (exp(2*n) * c^n * (2-c)^(n-1)), where c = -LambertW(-2*exp(-2)) = -A226775. - _Vaclav Kotesovec_, Aug 11 2021
%e E.g.f.: 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! + 686179134994911311*x^10/10! + 435599748089861536402*x^11/11! + 334122749226062422725277*x^12/12! +...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k/k! in exp(n^2*x) / A(x)^n 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 RELATED SERIES.
%e The logarithmic derivative of the e.g.f. appears to be an integer series:
%e A'(x)/A(x) = 2 + 3*x + 28*x^2 + 475*x^3 + 11556*x^4 + 362418*x^5 + 13820696*x^6 + 617990499*x^7 + 31613351140*x^8 + 1817581003238*x^9 + ... + A319146(n+1)*x^n + ...
%o (PARI) {a(n) = my(A=[1]); 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 ); H=A;A[n+1]}
%o for(n=0,21, print1(a(n)*n!,", "))
%Y Cf. A319146, A317343, A303060.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 18 2018
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