A326010 G.f. A(x) satisfies: 0 = Sum_{n>=1} n * ((1+x)^n - A(x))^n.

1, 1, 2, 20, 282, 5134, 112053, 2823119, 80202565, 2529045393, 87523776013, 3295995672161, 134155142687732, 5869278171065418, 274718037952537674, 13701118397652347442, 725505704889894172448, 40658992718689480518864, 2404662897766073643050293, 149692182669205551972626617, 9784886698908632846522031701

Offset: 0

Author

Paul D. Hanna, Jun 04 2019

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 20*x^3 + 282*x^4 + 5134*x^5 + 112053*x^6 + 2823119*x^7 + 80202565*x^8 + 2529045393*x^9 + 87523776013*x^10 + ...
such that
0 = ((1+x) - A(x)) + 2*((1+x)^2 - A(x))^2 + 3*((1+x)^3 - A(x))^3 + 4*((1+x)^4 - A(x))^4 + 5*((1+x)^5 - A(x))^5 + 6*((1+x)^6 - A(x))^6 + ...
The terms a(n) modulo 2 begin:
1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,
0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,1,1,1,1,1,1,0,0,
0,0,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,1,1,1,
0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,1,1,1,1,
1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,1,
0,0,0,0,1,1,1,1,0,0,1,1,1,1,1,1,0,0,1,1,1,1,0,0,
1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0,
1,1,0,0,0,0,0,0,0, ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Programs

PARI

{a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = polcoeff( sum(m=1,#A, m* ((1+x)^m - Ser(A))^m ), #A-1));A[n+1]}
for(n=0,25,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(1) 0 = Sum_{n>=1} n * ((1+x)^n - A(x))^n.

(2) A(x) = P(x)/Q(x) where

P(x) = Sum_{n>=0} n * (1+x)^(n^2) / (1 + (1+x)^n*A(x))^(n+2),

Q(x) = Sum_{n>=0} (1+x)^(n*(n+1)) / (1 + (1+x)^n*A(x))^(n+2).

(3) A'(x) = P(x)/Q(x) where

P(x) = Sum_{n>=0} (n+1)^3 * ((1+x)^(n+1) - A(x))^n * (1+x)^n,

Q(x) = Sum_{n>=0} (n+1)^2 * ((1+x)^(n+1) - A(x))^n.

a(n) ~ c * d^n * sqrt(n) * n!, where d = A317855 = 3.16108865386... and c = 0.102568345138... - Vaclav Kotesovec, Jun 05 2019