A340894 G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(2*n)/(1 - x*A(x)^n) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(2*n+1)).

1, 1, 3, 18, 126, 958, 7707, 64564, 557519, 4928784, 44398592, 406119900, 3762522607, 35236156779, 333052648607, 3173351871750, 30448930792460, 293980976342244, 2854070686906317, 27845872331417000, 272896298225616888, 2685318944226499424, 26521692874280137381

Offset

0,3

Comments

Equals row k = 2 of rectangular table A340940.

Links

Table of n, a(n) for n=0..22.

Formula

Given g.f. A(x), the following sums are all equal:

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

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

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

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

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

see the example section for the value of B(x).

Example

G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 126*x^4 + 958*x^5 + 7707*x^6 + 64564*x^7 + 557519*x^8 + 4928784*x^9 + 44398592*x^10 + 406119900*x^11 + ...
such that the following sums are all equal:
B(x) = 1/(1-x) + x*A(x)^2/(1 - x*A(x)) + x^2*A(x)^4/(1 - x*A(x)^2) + x^3*A(x)^6/(1 - x*A(x)^3) + x^4*A(x)^8/(1 - x*A(x)^4) + ...
and
B(x) = 1/(1-x*A(x)) + x*A(x)/(1 - x*A(x)^3) + x^2*A(x)^2/(1 - x*A(x)^5) + x^3*A(x)^3/(1 - x*A(x)^7) + x^4*A(x)^4/(1 - x*A(x)^9) + ...
also
B(x) = 1/(1-x*A(x)^2) + x/(1 - x*A(x)^3) + x^2/(1 - x*A(x)^4) + x^3/(1 - x*A(x)^5) + x^4/(1 - x*A(x)^6) + x^5/(1 - x*A(x)^7) + ...
where
B(x) = 1 + 2*x + 5*x^2 + 18*x^3 + 93*x^4 + 602*x^5 + 4406*x^6 + 34666*x^7 + 286098*x^8 + 2443548*x^9 + 21419500*x^10 + 191631430*x^11 + ...

Prog

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

Crossrefs

Cf. A340940, A340941, A340942, A340943, A340895.

Sequence in context: A176277 A289429 A004987 * A363560 A074557 A073971

Adjacent sequences: A340891 A340892 A340893 * A340895 A340896 A340897

Keyword

nonn

Author

Paul D. Hanna, Jan 26 2021

Status

approved