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%I #25 Oct 22 2020 03:32:05
%S 1,1,3,18,165,2019,30688,554784,11591649,274313325,7242994143,
%T 210931834662,6713206636084,231754182524900,8624280230971980,
%U 344124280164153056,14656294893872323449,663624782214112471329,31833832291287920426617,1612762327644980719082470,86050799297228500838101677,4823357354919905244973170883,283375597845431500054861239512
%N O.g.f. A(x) satisfies: [x^n] exp(-n*A(x)) / (1 - n*x) = 0, for n > 0.
%C It is remarkable that this sequence should consist entirely of integers.
%C Compare to: [x^n] exp(-n*G(x)) * (1 + n*x) = 0, for n > 0, when G(x) = x - x*G(x)*G'(x), where G(-x)/(-x) is the o.g.f. of A088716.
%H Paul D. Hanna, <a href="/A319938/b319938.txt">Table of n, a(n) for n = 1..300</a>
%F a(n) ~ c * n^(n-1), where c = 0.335949071234... - _Vaclav Kotesovec_, Oct 22 2020
%e O.g.f.: A(x) = x + x^2 + 3*x^3 + 18*x^4 + 165*x^5 + 2019*x^6 + 30688*x^7 + 554784*x^8 + 11591649*x^9 + 274313325*x^10 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k/k! in exp(-n*A(x)) / (1 - n*x) begins:
%e n=1: [1, 0, -1, -16, -423, -19616, -1444625, -154014624, ...];
%e n=2: [1, 0, 0, -20, -768, -38832, -2895680, -308705280, ...];
%e n=3: [1, 0, 3, 0, -783, -53568, -4309605, -465802704, ...];
%e n=4: [1, 0, 8, 56, 0, -50144, -5307200, -616050432, ...];
%e n=5: [1, 0, 15, 160, 2265, 0, -4729025, -711963600, ...];
%e n=6: [1, 0, 24, 324, 6912, 145584, 0, -613885824, ...];
%e n=7: [1, 0, 35, 560, 15057, 460768, 13696795, 0, ...];
%e n=8: [1, 0, 48, 880, 28032, 1050432, 44437120, 1769051136, 0, ...]; ...
%e in which the coefficient of x^n in row n forms a diagonal of zeros.
%e RELATED SERIES.
%e exp(A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 529*x^4/4! + 22581*x^5/5! + 1598011*x^6/6! + 166508413*x^7/7! + 23765885025*x^8/8! + ...
%e exp(-A(x)) = 1 - x - x^2/2! - 13*x^3/3! - 359*x^4/4! - 17501*x^5/5! - 1326929*x^6/6! - 143902249*x^7/7! - 21072159247*x^8/8! + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp(-m*x*Ser(A))/(1-m*x +x^2*O(x^m)))[m+1]/m ); A[n]}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A088716, A321085, A319939, A319940, A320417.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Oct 09 2018
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