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A304861 O.g.f. A(x) satisfies: 0 = [x^n] exp( n*(n-1) * Integral 1/A(x) dx ) / A(x), for n > 0.

%I A304861 #11 Oct 18 2020 08:16:35
%S A304861 1,0,2,20,328,7664,231744,8560512,372339840,18593869184,1046764673152,
%T A304861 65518908623360,4510397034460160,338534873778165760,
%U A304861 27505042556295458816,2404499023598887772160,225014884122460397678592,22441327480906466274779136,2376060993772932821157273600,266169866452350363506325897216,31451236460722731478509841711104
%N A304861 O.g.f. A(x) satisfies: 0 = [x^n] exp( n*(n-1) * Integral 1/A(x) dx ) / A(x), for n > 0.
%C A304861 Note: 0 = [x^n] exp( (n-1) * Integral 1/F(x) dx ) / F(x) holds for n > 0 when F(x) = sqrt(1 + x^2).
%C A304861 Note: 0 = [x^n] exp( n * Integral 1/G(x) dx ) / G(x) holds for n > 0 when G(x) = 1 + x.
%C A304861 It is remarkable that this sequence should consist entirely of integers.
%H A304861 Paul D. Hanna, <a href="/A304861/b304861.txt">Table of n, a(n) for n = 0..520</a>
%F A304861 a(n) ~ sqrt(1-c) * 2^(2*n - 2) * n^(n - 1/2) / (sqrt(Pi) * exp(n) * c^(n - 1/2) * (2-c)^(n-1)), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - _Vaclav Kotesovec_, Oct 18 2020
%e A304861 O.g.f.: A(x) = 1 + 2*x^2 + 20*x^3 + 328*x^4 + 7664*x^5 + 231744*x^6 + 8560512*x^7 + 372339840*x^8 + 18593869184*x^9 + 1046764673152*x^10 + ...
%e A304861 ILLUSTRATION OF DEFINITION.
%e A304861 The table of coefficients of x^k in  exp(n*(n-1) * Integral 1/A(x) dx)/A(x) begins:
%e A304861 n=0: [1, 0, -2, -20, -324, -7584, -230040, -8516976, ...];
%e A304861 n=1: [1, 0, -2, -20, -324, -7584, -230040, -8516976, ...];
%e A304861 n=2: [1, 2, 0, -24, -380, -8424, -248640, -9062720, ...];
%e A304861 n=3: [1, 6, 16, 0, -480, -10528, -292544, -10293696, ...];
%e A304861 n=4: [1, 12, 70, 236, 0, -13472, -378336, -12576960, ...];
%e A304861 n=5: [1, 20, 198, 1260, 5176, 0, -485520, -16616864, ...];
%e A304861 n=6: [1, 30, 448, 4400, 31176, 151792, 0, -21316608, ...];
%e A304861 n=7: [1, 42, 880, 12216, 125340, 989384, 5588416, 0, ...]; ...
%e A304861 in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp( n*(n-1) * Integral 1/A(x) dx ) / A(x), for n > 0.
%e A304861 RELATED SERIES.
%e A304861 1/A(x) = 1 - 2*x^2 - 20*x^3 - 324*x^4 - 7584*x^5 - 230040*x^6 - 8516976*x^7 - 371005040*x^8 - 18545507840*x^9 - 1044727771680*x^10 + ...
%e A304861 exp(Integral 1/A(x) dx) = 1 + 2*x/2 + 2*x^2/2^2 - 4*x^3/2^3 - 90*x^4/2^4 - 2244*x^5/2^5 - 85196*x^6/2^6 - 4372040*x^7/2^7 - 281105594*x^8/2^8 - 21659046420*x^9/2^9 + ...
%e A304861 exp(2 * Integral 1/A(x) dx) = 1 + 2*x + 2*x^2 - 12*x^4 - 152*x^5 - 2808*x^6 - 71040*x^7 - 2265680*x^8 - 86833824*x^9 - 3878209440*x^10 - 197532405760*x^11 + ..., an integer series.
%e A304861 A'(x)/A(x) = 4*x + 60*x^2 + 1304*x^3 + 38120*x^4 + 1385344*x^5 + 59770928*x^6 + 2973371104*x^7 + 167126930016*x^8 + 10457452841984*x^9 + ...
%o A304861 (PARI) {a(n) = my(A=[1],m); for(i=1,n+1, m=#A; A=concat(A,0); A[m+1] = Vec( exp(m*(m-1)*intformal(1/Ser(A))) / Ser(A) )[m+1] );A[n+1]}
%o A304861 for(n=0,20,print1(a(n),", "))
%Y A304861 Cf. A304862, A305144, A305145, A305146, A305147.
%Y A304861 Cf. A305137, A305138, A305139, A305140, A305141, A305142, A305143.
%K A304861 nonn
%O A304861 0,3
%A A304861 _Paul D. Hanna_, Jun 01 2018