login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 05:22 EST 2022. Contains 358594 sequences. (Running on oeis4.)