A341342 G.f. A(x) satisfies: A(x) = P(x)/Q(x) where P(x) = Sum_{n>=0} (n+1)*x^n/(1 - x*A(x)^n) and Q(x) = Sum_{n>=0} x^n/(1 - x*A(x)^n).

1, 1, 1, 2, 5, 15, 49, 170, 616, 2313, 8948, 35518, 144208, 597489, 2521733, 10827165, 47243223, 209338510, 941475437, 4295923404, 19883249375, 93334907842, 444331270293, 2145284774617, 10505316792339, 52182498814037, 262957463261285, 1344455358497182

Offset

0,4

Links

Table of n, a(n) for n=0..27.

Formula

G.f. A(x) satisfies:

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

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

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

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

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

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

Example

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 49*x^6 + 170*x^7 + 616*x^8 + 2313*x^9 + 8948*x^10 + 35518*x^11 + 144208*x^12 + 597489*x^13 + ...
such that A(x) = P(x)/Q(x) where
P(x) = 1/(1-x) + 2*x/(1 - x*A(x)) + 3*x^2/(1 - x*A(x)^2) + 4*x^3/(1 - x*A(x)^3) + 5*x^4/(1 - x*A(x)^4) + 6*x^5/(1 - x*A(x)^5) + ...
Q(x) = 1/(1-x) + x/(1 - x*A(x)) + x^2/(1 - x*A(x)^2) + x^3/(1 - x*A(x)^3) + x^4/(1 - x*A(x)^4) + x^5/(1 - x*A(x)^5) + ...
also
P(x) = 1/(1-x)^2 + x/(1 - x*A(x))^2 + x^2/(1 - x*A(x)^2)^2 + x^3/(1 - x*A(x)^3)^2 + x^4/(1 - x*A(x)^4)^2 + x^5/(1 - x*A(x)^5)^2 + ...
explicitly,
P(x) = 1 + 3*x + 6*x^2 + 12*x^3 + 27*x^4 + 70*x^5 + 204*x^6 + 648*x^7 + 2192*x^8 + 7780*x^9 + 28696*x^10 + 109278*x^11 + 427652*x^12 + ...
Q(x) = 1 + 2*x + 3*x^2 + 5*x^3 + 10*x^4 + 24*x^5 + 66*x^6 + 200*x^7 + 650*x^8 + 2228*x^9 + 7966*x^10 + 29485*x^11 + 112368*x^12 + ...

Prog

(PARI) {a(n) = my(A=1+x+x*O(x^n), P=1, Q=1);
for(i=0, n,
P = sum(m=0, n, (m+1)*x^m/(1 - x*A^m + x*O(x^n)) );
Q = sum(m=0, n, x^m/(1 - x*A^m + x*O(x^n)) );
A = P/Q); polcoeff(H=A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1+x+x*O(x^n), P=1, Q=1);
for(i=0, n,
P = sum(m=0, n, x^m/(1 - x*A^m + x*O(x^n))^2 );
Q = sum(m=0, n, x^m/(1 - x*A^m + x*O(x^n)) );
A = P/Q); polcoeff(H=A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A340942, A341382.

Sequence in context: A084082 A190270 A364588 * A149939 A149940 A149941

Adjacent sequences: A341339 A341340 A341341 * A341343 A341344 A341345

Keyword

nonn

Author

Paul D. Hanna, Feb 09 2021

Status

approved