A306067 - OEIS

%I #12 Jul 12 2018 03:45:22

%S 1,4,21,178,2279,39066,835132,21400198,640239525,21920851282,

%T 845615003996,36298192983482,1716348366690487,88653661788525666,

%U 4967006270867149524,300043327305644202366,19440451816128996788777,1344909407655243937857826,98949254253416815493778796,7714902418308597200477578514,635444724815621395463510504211,55134789286331454820101232131938

%N E.g.f. A(x) satisfies:  Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n-k) + (k+1)*A(x)  =  1.

%H Vaclav Kotesovec, <a href="/A306067/b306067.txt">Table of n, a(n) for n = 0..128</a> (terms 0..100 from Paul D. Hanna)

%F E.g.f. A(x) satisfies:

%F (1) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n-k) + k*A(x)  =  -x.

%F (2) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n-k) + (k+1)*A(x)  =  1.

%F (3) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n-k+1) + k*A(x)  =  1/A(x).

%F a(n)/n! ~ c * d^n / n^(3/2), where d = 4.423034555284689... and c = 3.17922741818... - _Vaclav Kotesovec_, Jul 12 2018

%e G.f.: A(x) = 1 + 4*x + 21*x^2/2! + 178*x^3/3! + 2279*x^4/4! + 39066*x^5/5! + 835132*x^6/6! + 21400198*x^7/7! + 640239525*x^8/8! + 21920851282*x^9/9! + 845615003996*x^10/10! + ...

%e such that

%e 1  =  (0 + A(x))  -  (1 + A(x))*(0 + 2*A(x))*x  +  (2 + A(x))*(1 + 2*A(x))*(0 + 3*A(x))*x^2/2!  -  (3 + A(x))*(2 + 2*A(x))*(1 + 3*A(x))*(0 + 4*A(x))*x^3/3!  +  (4 + A(x))*(3 + 2*A(x))*(2 + 3*A(x))*(1 + 4*A(x))*(0 + 5*A(x))*x^4/4!  -  (5 + A(x))*(4 + 2*A(x))*(3 + 3*A(x))*(2 + 4*A(x))*(1 + 5*A(x))*(0 + 6*A(x))*x^5/5!  +  ...

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = -Vec( sum(n=0,#A, (-x)^n/n!* prod(k=0,n, (n-k) + (k+1)*Ser(A) ) ) )[#A] ); n!*A[n+1]}

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

%Y Cf. A306090.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 25 2018