A322736 G.f. satisfies: A(x) = Sum_{n>=0} 2^n * ( (1+x)^n - A(x)^(1/2) )^n / ( 3 - 2*(1+x)^n * A(x)^(1/2) )^(n+1).

1, 2, 8, 96, 2956, 114992, 5244896, 277303392, 16680895688, 1124043943848, 83860544099264, 6863636560150656, 611673708807594944, 58982083391411043456, 6120766911879901270784, 680339106407429897733760, 80661483112436517009089168, 10162784535291704640507410016, 1356175692780348173552997926272, 191103836643650458447321745220736, 28358934286111202643351952170366400, 4420810085328675478052952299755080000

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

G.f. A(x) and B(x) = A(x)^(1/2) satisfy:

(1) A(x) = Sum_{n>=0} 2^n * ( (1+x)^n - B(x) )^n / ( 3 - 2*(1+x)^n * B(x) )^(n+1),

(2) A(x) = Sum_{n>=0} 2^n * ( (1+x)^n + B(x) )^n / ( 3 + 2*(1+x)^n * B(x) )^(n+1).

Example

G.f.: A(x) = 1 + 2*x + 8*x^2 + 96*x^3 + 2956*x^4 + 114992*x^5 + 5244896*x^6 + 277303392*x^7 + 16680895688*x^8 + 1124043943848*x^9 + ...
such that A(x) and B = A(x)^(1/2) satisfy
A(x) = 1/(3 - 2*B)  +  2*((1+x) - B)/(3 - 2*(1+x)*B)^2  +  2^2*((1+x)^2 - B)^2/(3 - 2*(1+x)^2*B)^3  +  2^3*((1+x)^3 - B)^3/(3 - 2*(1+x)^3*B)^4  +  2^4*((1+x)^4 - B)^4/(3 - 2*(1+x)^4*B)^5  +  2^5*((1+x)^5 - B)^5/(3 - 2*(1+x)^5*B)^6 + ...
also,
A(x) = 1/(3 + 2*B)  +  2*((1+x) + B)/(3 + 2*(1+x)*B)^2  +  2^2*((1+x)^2 + B)^2/(3 + 2*(1+x)^2*B)^3  +  2^3*((1+x)^3 + B)^3/(3 + 2*(1+x)^3*B)^4  +  2^4*((1+x)^4 + B)^4/(3 + 2*(1+x)^4*B)^5  +  2^5*((1+x)^5 + B)^5/(3 + 2*(1+x)^5*B)^6 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( (1+x)^m - Ser(A)^(1/2) )^m *2^m / (3 - 2*(1+x)^m*Ser(A)^(1/2) )^(m+1) ) ) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A317350, A322735.

Sequence in context: A255132 A297332 A009752 * A137704 A001417 A156926

Adjacent sequences: A322733 A322734 A322735 * A322737 A322738 A322739

Keyword

nonn

Author

Paul D. Hanna, Jan 25 2019

Status

approved