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A337579 O.g.f. A(x) satisfies: [x^n] exp( n*(n+1)^2 * x ) / A(x)^((n+1)^2) = 0 for n>0.

%I A337579 #5 Sep 02 2020 04:49:33
%S A337579 1,1,5,202,25741,6481768,2661785172,1606979708104,1336018641201031,
%T A337579 1461946920710738032,2036450966030220362632,3519187269661662800713808,
%U A337579 7390652429852470066011519746,18545709306030399397877283499248,54823008100459892066683079355901888,188621839026471088419358039473633535392
%N A337579 O.g.f. A(x) satisfies: [x^n] exp( n*(n+1)^2 * x ) / A(x)^((n+1)^2) = 0 for n>0.
%C A337579 It is remarkable that this sequence consists entirely of integers.
%e A337579 O.g.f.: A(x) = 1 + x + 5*x^2 + 202*x^3 + 25741*x^4 + 6481768*x^5 + 2661785172*x^6 + 1606979708104*x^7 + 1336018641201031*x^8 + ...
%e A337579 ILLUSTRATION OF DEFINITION.
%e A337579 The table of coefficients of x^k/k! in exp(n*(n+1)^2*x) / A(x)^((n+1)^2) begins:
%e A337579 n=0: [1, -1, -8, -1158, -607824, -771471360, -1906996245120, ...];
%e A337579 n=1: [1, 0, -36, -4736, -2447112, -3096809856, -7645376634080, ...];
%e A337579 n=2: [1, 9, 0, -12114, -5911488, -7219467792, -17580593299968, ...];
%e A337579 n=3: [1, 32, 880, 0, -12002784, -14133084672, -33100636472064, ...];
%e A337579 n=4: [1, 75, 5400, 341650, 0, -25227867600, -57875848640000, ...];
%e A337579 n=5: [1, 144, 20412, 2803392, 343375416, 0, -95154559008480, ...];
%e A337579 n=6: [1, 245, 59584, 14323974, 3357877488, 709290480864, 0, ...];
%e A337579 n=7: [1, 384, 146880, 55883776, 21079051392, 7789007628288, 2612787154865152, 0, ...]; ...
%e A337579 in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp(n*(n+1)^2*x) / A(x)^((n+1)^2) = 0 for n>0.
%e A337579 RELATED SERIES.
%e A337579 log(A(x)) = x + 9*x^2/2 + 592*x^3/3 + 102125*x^4/4 + 32276196*x^5/5 + 15931091190*x^6/6 + 11230009495552*x^7/7 + 10675195543084221*x^8/8 + ...
%e A337579 where [x^n] exp( (n+1)^2 * (n*x - log(A(x))) ) = 0 for n>0.
%o A337579 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = polcoeff( exp(m^2*(m-1)*x +x*O(x^m)) / Ser(A)^(m^2), m-1) /m^2 );A[n+1]}
%o A337579 for(n=0, 20, print1(a(n), ", "))
%Y A337579 Cf. A337458.
%K A337579 nonn
%O A337579 0,3
%A A337579 _Paul D. Hanna_, Sep 01 2020