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%I #8 Sep 30 2018 19:41:43
%S 1,7,495,79066,23024860,10572117273,7013015135896,6339726763590496,
%T 7490028944758474548,11205332666103104972870,
%U 20707904898272234027832128,46337521327783545461807594898,123493236514176445337375562127273,386578735289710248386597916624905008,1404667043036383301054968922912928274395,5864184572831214357534720189729386105896368
%N O.g.f. A(x) satisfies: [x^n] (1+x)^(n^3) / exp( n*A(x) ) = 0 for n >= 1.
%C It is remarkable that this sequence should consist entirely of integers.
%e G.f.: A(x) = x + 7*x^2 + 495*x^3 + 79066*x^4 + 23024860*x^5 + 10572117273*x^6 + 7013015135896*x^7 + 6339726763590496*x^8 + ...
%e The table of coefficients of x^k/k! in (1+x)^(n^3)/exp(n*A(x)) begins
%e n=1: [1, 0, -15, -2968, -1896915, -2762537976, -7611409439315, ...];
%e n=2: [1, 6, 0, -6356, -3939984, -5639776032, -15422380789952, ...];
%e n=3: [1, 24, 507, 0, -6435495, -9016860528, -24073494668037, ...];
%e n=4: [1, 60, 3480, 182648, 0, -13206756864, -34841304393920, ...];
%e n=5: [1, 120, 14205, 1643200, 174129405, 0, -48106118941175, ...];
%e n=6: [1, 210, 43800, 9054612, 1839707280, 344760621984, 0, ...];
%e n=7: [1, 336, 112455, 37468424, 12407062521, 4051917824808, 1225838213682103, 0, ...]; ...
%e in which the coefficient of x^n in the n-th row forms a diagonal of zeros.
%e RELATED SERIES.
%e exp(A(x)) = 1 + x + 15*x^2/2! + 3013*x^3/3! + 1910137*x^4/4! + 2772919701*x^5/5! + 7629020105551*x^6/6! + 35399950893130825*x^7/7! + ...
%o (PARI) {a(n) = my(A=[1], m); for(i=1, n+1, A=concat(A, 0); m=#A; A[m] = Vec( (1+x +x^2*O(x^m))^(m^3) * exp(-m*x*Ser(A)) )[m+1]/m ); polcoeff( x*Ser(A), n)}
%o for(n=1, 20, print1(a(n), ", "))
%Y Cf. A319832, A319833.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Sep 28 2018
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