A322735 G.f. satisfies: A(x) = Sum_{n>=0} ( (1+x)^n - A(x)^(1/2) )^n / ( 2 - (1+x)^n * A(x)^(1/2) )^(n+1).

1, 1, 4, 32, 424, 7696, 173442, 4619266, 141315896, 4874012942, 186981188532, 7896318230898, 364045464940596, 18196879341802488, 980406767669688312, 56648325010279262864, 3494752526532046751322, 229295129566323954429582, 15944415062268028208782178, 1171388932048172852048806000, 90667183883120180538001042398

Offset

0,3

Comments

It is remarkable that the g.f. should consist entirely of integer coefficients.

Links

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

Formula

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

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

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

Example

G.f.: A(x) = 1 + x + 4*x^2 + 32*x^3 + 424*x^4 + 7696*x^5 + 173442*x^6 + 4619266*x^7 + 141315896*x^8 + 4874012942*x^9 + 186981188532*x^10 + ...
such that A(x) and B = A(x)^(1/2) satisfy
A(x) = 1/(2 - B)  +  ((1+x) - B)/(2 - (1+x)*B)^2  +  ((1+x)^2 - B)^2/(2 - (1+x)^2*B)^3  +  ((1+x)^3 - B)^3/(2 - (1+x)^3*B)^4  +  ((1+x)^4 - B)^4/(2 - (1+x)^4*B)^5  +  ((1+x)^5 - B)^5/(2 - (1+x)^5*B)^6 + ...
also,
A(x) = 1/(2 + B)  +  ((1+x) + B)/(2 + (1+x)*B)^2  +  ((1+x)^2 + B)^2/(2 + (1+x)^2*B)^3  +  ((1+x)^3 + B)^3/(2 + (1+x)^3*B)^4  +  ((1+x)^4 + B)^4/(2 + (1+x)^4*B)^5  +  ((1+x)^5 + B)^5/(2 + (1+x)^5*B)^6 + ...
Notice that A(x)^(1/2) is not an integer series, but instead begins
A(x)^(1/2) = 1 + 2*(x/4) + 30*(x/4)^2 + 964*(x/4)^3 + 51894*(x/4)^4 + 3807644*(x/4)^5 + 345572460*(x/4)^6 + 36985627016*(x/4)^7 + 4541283789862*(x/4)^8 + 628123762214444*(x/4)^9 + 96578670976842436*(x/4)^10 + ...
thus, given the definition, it is remarkable that A(x) should be an integer series.

Prog

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

Crossrefs

Cf. A317350, A322737.

Sequence in context: A005172 A298694 A222685 * A140178 A366685 A088991

Adjacent sequences: A322732 A322733 A322734 * A322736 A322737 A322738

Keyword

nonn

Author

Paul D. Hanna, Jan 24 2019

Status

approved