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%I #10 Oct 19 2018 17:52:58
%S 1,4,243,90624,85137125,166983141708,584752700234290,
%T 3335932982893551104,28979545952901285126801,
%U 364345886028800419659490500,6369639791888600743755572216796,149926538998807813901526056378836416,4626572216398455689960837772846170271886,183057653659252604698726467223480475509456616,9112803025595308606953230928489236750492759413500
%N O.g.f. A(x) satisfies: [x^n] exp(-n^3*A(x)) / (1 - n^3*x) = 0, for n > 0.
%C It is remarkable that this sequence should consist entirely of integers.
%e O.g.f.: A(x) = x + 4*x^2 + 243*x^3 + 90624*x^4 + 85137125*x^5 + 166983141708*x^6 + 584752700234290*x^7 + 3335932982893551104*x^8 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k/k! in exp(-n^3*A(x)) / (1 - n^3*x) begins:
%e n=1: [1, 0, -7, -1456, -2174823, -10216353056, -120227612463575, ...];
%e n=2: [1, 0, 0, -10640, -17375232, -81730853568, -961821732684800, ...];
%e n=3: [1, 0, 513, 0, -54746199, -275499911232, -3246531106517055, ...];
%e n=4: [1, 0, 3584, 430976, 0, -612637136384, -7685860529991680, ...];
%e n=5: [1, 0, 14625, 3724000, 1834643625, 0, -14123406888329375, ...];
%e n=6: [1, 0, 44928, 19840464, 18646474752, 17991609015744, 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 + 9*x^2/2! + 1483*x^3/3! + 2181049*x^4/4! + 10227462141*x^5/5! + 120289476378841*x^6/6! + 2947997248178316559*x^7/7! + ...
%e exp(-A(x)) = 1 - x - 7*x^2/2! - 1435*x^3/3! - 2168999*x^4/4! - 10205478941*x^5/5! - 120166314345239*x^6/6! - 2946310403245714303*x^7/7! + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(-m^3*x*Ser(A))/(1-m^3*x +x^2*O(x^m)))[m+1]/m^3 ); A[n]}
%o for(n=1, 20, print1(a(n), ", "))
%Y Cf. A320417, A319938, A319939, A320668, A320669.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Oct 15 2018
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