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A319939 O.g.f. A(x) satisfies: [x^n] exp(-n^2*A(x)) / (1 - n*x)^n = 0, for n > 0.

%I A319939 #16 Oct 24 2020 03:10:08
%S A319939 1,1,3,24,325,6642,176204,5828160,228372291,10374419250,534203188948,
%T A319939 30762752950224,1956914341159778,136286437739608492,
%U A319939 10310240639621093400,841935232438747348480,73807352585103519962815,6913603998931859925828282,689148541231545351838902508,72838943589708142133363904400,8137053663063956034586144506558,958035702236154579666369909892724
%N A319939 O.g.f. A(x) satisfies: [x^n] exp(-n^2*A(x)) / (1 - n*x)^n = 0, for n > 0.
%C A319939 It is remarkable that this sequence should consist entirely of integers.
%H A319939 Paul D. Hanna, <a href="/A319939/b319939.txt">Table of n, a(n) for n = 1..200</a>
%F A319939 a(n) ~ c * d^n * n! / n^3, where d = 6.1103392... and c = 0.05165... - _Vaclav Kotesovec_, Oct 24 2020
%e A319939 O.g.f.: A(x) = x + x^2 + 3*x^3 + 24*x^4 + 325*x^5 + 6642*x^6 + 176204*x^7 + 5828160*x^8 + 228372291*x^9 + 10374419250*x^10 + ...
%e A319939 ILLUSTRATION OF DEFINITION.
%e A319939 The table of coefficients of x^k/k! in exp(-n^2*A(x)) / (1 - n*x)^n begins:
%e A319939 n=1: [1, 0, -1, -16, -567, -38816, -4771025, -886931424, ...];
%e A319939 n=2: [1, 0, 0, -40, -2112, -154464, -19097600, -3549131520, ...];
%e A319939 n=3: [1, 0, 9, 0, -3483, -333504, -43269795, -8050921776, ...];
%e A319939 n=4: [1, 0, 32, 224, 0, -454016, -75031040, -14515172352, ...];
%e A319939 n=5: [1, 0, 75, 800, 21225, 0, -92559125, -22271154000, ...];
%e A319939 n=6: [1, 0, 144, 1944, 88128, 2515104, 0, -25624491264, ...];
%e A319939 n=7: [1, 0, 245, 3920, 252693, 10516576, 505622425, 0, ...];
%e A319939 n=8: [1, 0, 384, 7040, 602112, 30829056, 2210682880, 134210187264, 0, ...];
%e A319939 in which the coefficient of x^n in row n forms a diagonal of zeros.
%e A319939 RELATED SERIES.
%e A319939 exp(A(x)) = 1 + x + 3*x^2 + 25*x^3/3! + 673*x^4/4! + 42501*x^5/5! + 5048251*x^6/6! + 924544573*x^7/7! + 242568147585*x^8/8! + ...
%o A319939 (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp(-m^2*x*Ser(A))/(1-m*x +x^2*O(x^m))^m)[m+1]/m^2 ); A[n]}
%o A319939 for(n=1,30,print1(a(n),", "))
%Y A319939 Cf. A319938, A319940, A320669.
%K A319939 nonn
%O A319939 1,3
%A A319939 _Paul D. Hanna_, Oct 09 2018