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A340358 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (n+1) * x^n / (1 - x^(n+1)*A(x))^3. 3
1, 5, 24, 152, 1094, 8508, 69565, 588469, 5106516, 45199827, 406485567, 3703483221, 34111556603, 317103532465, 2971283282979, 28033510000286, 266092385194061, 2539244496436404, 24346664830510834, 234435203932318053, 2266062742515203697, 21980115620177318458 (list; graph; refs; listen; history; text; internal format)
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 = 2, and t = 3.

LINKS

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

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))^3.

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

EXAMPLE

G.f.: A(x) = 1 + 5*x + 24*x^2 + 152*x^3 + 1094*x^4 + 8508*x^5 + 69565*x^6 + 588469*x^7 + 5106516*x^8 + 45199827*x^9 + 406485567*x^10 + ...

where

A(x) = 1/(1 - x*A(x))^3 + 2*x/(1 - x^2*A(x))^3 + 3*x^2/(1 - x^3*A(x))^3 + 4*x^3/(1 - x^4*A(x))^3 + 5*x^4/(1 - x^5*A(x))^3 + ...

also

A(x) = 1/(1 - x)^2 + 3*x*A(x)/(1 - x^2)^2 + 6*x^2*A(x)^2/(1 - x^3)^2 + 10*x^3*A(x)^3/(1 - x^4)^2 + 15*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))^3 )); 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, (m+1)*(m+2)/2 * 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, A340357, A340359, A340360.

Sequence in context: A228067 A322208 A241134 * A009424 A184597 A009601

Adjacent sequences:  A340355 A340356 A340357 * A340359 A340360 A340361

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 07 2021

STATUS

approved

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Last modified May 13 01:58 EDT 2021. Contains 343830 sequences. (Running on oeis4.)