A305108 - OEIS

%I #5 Jun 06 2018 21:33:42

%S 1,1,12,342,16080,1090400,99884736,11885278104,1786708056832,

%T 331931252093472,74805826012157600,20127855750577630968,

%U 6377560491906482613120,2351353212746078192866032,998307775668524681354287776,483648245019895895020555792200,265226237170176555086800587134976,163463812810277012465203148994919744,112505648337664454361768261713783693856

%N O.g.f. A(x) satisfies: [x^n] exp( n^2 * x*A(x)^2 ) * (2 - A(x)) = 0 for n > 0.

%C Note: given F(x) = 1 + x * d/dx x*F(x)^2, where x*F(x) is a g.f. of A000699, then

%C (1) [x^n] exp( x*F(x)^2 ) * (n + 1 - F(x)) = 0 for n > 0,

%C (2) [x^n] exp( n * x*F(x)^2 ) * (2 - F(x)) = 0 for n > 0.

%C It is remarkable that this sequence should consist entirely of integers.

%e O.g.f.: A(x) = 1 + x + 12*x^2 + 342*x^3 + 16080*x^4 + 1090400*x^5 + 99884736*x^6 + 11885278104*x^7 + 1786708056832*x^8 + 331931252093472*x^9 + ...

%e RELATED SERIES.

%e A(x)^2 = 1 + 2*x + 25*x^2 + 708*x^3 + 32988*x^4 + 2221168*x^5 + 202453156*x^6 + 24007494000*x^7 + 3600588303536*x^8 + 667824555398528*x^9 + ...

%e exp(x*A(x)^2) = 1 + x + 5*x^2/2! + 163*x^3/3! + 17665*x^4/4! + 4051301*x^5/5! + 1624532461*x^6/6! + 1032073147855*x^7/7! + 976416036692993*x^8/8! + ...

%o (PARI) {a(n) = my(A=[1],m); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp( (m-1)^2*x*(Ser(A)^2) ) * (2 - Ser(A)) )[m] );A[n+1]}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A305107.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 27 2018