A340357 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (n+1) * x^n / (1 - x^(n+1)*A(x)).

1, 3, 7, 20, 64, 232, 908, 3717, 15716, 67996, 299396, 1337022, 6040421, 27556567, 126762966, 587324586, 2738338960, 12837950292, 60483207417, 286206067039, 1359678614745, 6482510515788, 31006901328525, 148750651958227, 715545729962692, 3450638733403489

Offset

0,2

Comments

The g.f. A(x) of this sequence is motivated by the following identity:

Sum_{n>=0} C(t+n-1,n) * p^n/(1 - q*r^n)^s = Sum_{n>=0} C(s+n-1,n) * q^n/(1 - p*r^n)^t ;

here, p = x, q = x*A(x), r = x, s = 1, and t = 2.

Links

Table of n, a(n) for n=0..25.

Formula

G.f. A(x) satisfies the following relations.

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

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

Example

G.f.: A(x) = 1 + 3*x + 7*x^2 + 20*x^3 + 64*x^4 + 232*x^5 + 908*x^6 + 3717*x^7 + 15716*x^8 + 67996*x^9 + 299396*x^10 + ...
where
A(x) = 1/(1 - x*A(x)) + 2*x/(1 - x^2*A(x)) + 3*x^2/(1 - x^3*A(x)) + 4*x^3/(1 - x^4*A(x)) + 5*x^4/(1 - x^5*A(x)) + ...
also
A(x) = 1/(1 - x)^2 + x*A(x)/(1 - x^2)^2 + x^2*A(x)^2/(1 - x^3)^2 + x^3*A(x)^3/(1 - x^4)^2 + x^4*A(x)^4/(1 - x^5)^2 + ...

Prog

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

Crossrefs

Cf. A340329, A340338, A340355, A340356, A340358, A340359, A340360.

Sequence in context: A320740 A320741 A292503 * A071688 A232687 A211602

Adjacent sequences: A340354 A340355 A340356 * A340358 A340359 A340360

Keyword

nonn

Author

Paul D. Hanna, Jan 07 2021

Status

approved