A305107 - OEIS

%I #8 Feb 22 2024 20:34:09

%S 1,1,10,216,7852,427770,32649276,3333409849,439648389640,

%T 72863444853189,14835946021507520,3642615447410904525,

%U 1061681881255681884336,362470236144441144939674,143310411318629778406908494,64968204494588611586367685020,33478892881907679134025607700400,19460912067689653469231875029090451,12674293598137775224869798728198782626,9192057681791476282831341020711249418814

%N O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x)^2 ) * (n + 1 - 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 + 10*x^2 + 216*x^3 + 7852*x^4 + 427770*x^5 + 32649276*x^6 + 3333409849*x^7 + 439648389640*x^8 + 72863444853189*x^9 + ...

%e RELATED SERIES.

%e A(x)^2 = 1 + 2*x + 21*x^2 + 452*x^3 + 16236*x^4 + 875564*x^5 + 66357788*x^6 + 6744065714*x^7 + 886863035042*x^8 + 146693676869950*x^9 + ...

%e exp(x*A(x)^2) = 1 + x + 5*x^2/2! + 139*x^3/3! + 11425*x^4/4! + 2009141*x^5/5! + 643102861*x^6/6! + 339114884935*x^7/7! + 274704279360449*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)*x*(Ser(A)^2) ) * (m - Ser(A)) )[m] );A[n+1]}

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

%Y Cf. A305108.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 27 2018