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 A206638 G.f. satisfies: A(x) = Sum_{n>=0} 3^n*A(x)^n * x^(n^2) / Product_{k=1..n} (1 - 3*x^k)*(1 - x^k*A(x)). 1
 1, 3, 21, 147, 1074, 8076, 62454, 494292, 3990378, 32756142, 272715870, 2297982828, 19563641319, 168036314862, 1454458825605, 12674387617266, 111104771086812, 979101922849230, 8668964794053837, 77080072176742422, 687976906966730076, 6161811541538326680 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA G.f. satisfies the identities: (1) A(x) = 1 + Sum_{n>=1} 3*x^n*A(x)^n / Product_{k=1..n} (1 - 3*x^k). (2) A(x) = 1 + Sum_{n>=1} 3^n*x^n*A(x) / Product_{k=1..n} (1 - x^k*A(x)). EXAMPLE G.f.: A(x) = 1 + 3*x + 21*x^2 + 147*x^3 + 1074*x^4 + 8076*x^5 +... where the g.f. satisfies: (0) A(x) = 1 + 3*x*A(x)/((1-3*x)*(1-x*A(x))) + 9*x^4*A(x)^2/((1-3*x)*(1-3*x^2)*(1-x*A(x))*(1-x^2*A(x))) + 27*x^9*A(x)^3/((1-3*x)*(1-3*x^2)*(1-3*x^3)*(1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +... (1) A(x) = 1 + 3*x*A(x)/(1-3*x) + 3*x^2*A(x)^2/((1-3*x)*(1-3*x^2)) + 3*x^3*A(x)^3/((1-3*x)*(1-3*x^2)*(1-3*x^3)) +... (2) A(x) = 1 + 3*x*A(x)/(1-x*A(x)) + 9*x^2*A(x)/((1-x*A(x))*(1-x^2*A(x))) + 27*x^3*A(x)/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +... PROG (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*3^m*A^m/prod(k=1, m, (1-3*x^k)*(1-x^k*A+x*O(x^n))))); polcoeff(A, n)} (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, 3*x^m*A^m/prod(k=1, m, (1-3*x^k+x*O(x^n))))); polcoeff(A, n)} (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, 3^m*x^m*A/prod(k=1, m, (1-x^k*A+x*O(x^n))))); polcoeff(A, n)} for(n=0, 35, print1(a(n), ", ")) CROSSREFS Cf. A145268, A196150, A206637. Sequence in context: A323203 A169634 A088088 * A037761 A037649 A037768 Adjacent sequences:  A206635 A206636 A206637 * A206639 A206640 A206641 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 10 2012 STATUS approved

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Last modified July 24 18:34 EDT 2021. Contains 346273 sequences. (Running on oeis4.)
The text passage is a detailed description of a specific integer sequence (A206638) from the On-Line Encyclopedia of Integer Sequences (OEIS). It includes the generating function, formulas, examples, and PARI code for computing the sequence.
no_question

the passage does not contain any explicit mathematical questions or word problems that require solving. it is purely a descriptive entry about a particular integer sequence.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A206638 G.f. satisfies: A(x) = Sum_{n>=0} 3^n*A(x)^n * x^(n^2) / Product_{k=1..n} (1 - 3*x^k)*(1 - x^k*A(x)). 1
 1, 3, 21, 147, 1074, 8076, 62454, 494292, 3990378, 32756142, 272715870, 2297982828, 19563641319, 168036314862, 1454458825605, 12674387617266, 111104771086812, 979101922849230, 8668964794053837, 77080072176742422, 687976906966730076, 6161811541538326680 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA G.f. satisfies the identities: (1) A(x) = 1 + Sum_{n>=1} 3*x^n*A(x)^n / Product_{k=1..n} (1 - 3*x^k). (2) A(x) = 1 + Sum_{n>=1} 3^n*x^n*A(x) / Product_{k=1..n} (1 - x^k*A(x)). EXAMPLE G.f.: A(x) = 1 + 3*x + 21*x^2 + 147*x^3 + 1074*x^4 + 8076*x^5 +... where the g.f. satisfies: (0) A(x) = 1 + 3*x*A(x)/((1-3*x)*(1-x*A(x))) + 9*x^4*A(x)^2/((1-3*x)*(1-3*x^2)*(1-x*A(x))*(1-x^2*A(x))) + 27*x^9*A(x)^3/((1-3*x)*(1-3*x^2)*(1-3*x^3)*(1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +... (1) A(x) = 1 + 3*x*A(x)/(1-3*x) + 3*x^2*A(x)^2/((1-3*x)*(1-3*x^2)) + 3*x^3*A(x)^3/((1-3*x)*(1-3*x^2)*(1-3*x^3)) +... (2) A(x) = 1 + 3*x*A(x)/(1-x*A(x)) + 9*x^2*A(x)/((1-x*A(x))*(1-x^2*A(x))) + 27*x^3*A(x)/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +... PROG (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*3^m*A^m/prod(k=1, m, (1-3*x^k)*(1-x^k*A+x*O(x^n))))); polcoeff(A, n)} (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, 3*x^m*A^m/prod(k=1, m, (1-3*x^k+x*O(x^n))))); polcoeff(A, n)} (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, 3^m*x^m*A/prod(k=1, m, (1-x^k*A+x*O(x^n))))); polcoeff(A, n)} for(n=0, 35, print1(a(n), ", ")) CROSSREFS Cf. A145268, A196150, A206637. Sequence in context: A323203 A169634 A088088 * A037761 A037649 A037768 Adjacent sequences:  A206635 A206636 A206637 * A206639 A206640 A206641 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 10 2012 STATUS approved

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Last modified July 24 18:34 EDT 2021. Contains 346273 sequences. (Running on oeis4.)
user
The text passage is a detailed description of a specific integer sequence (A206638) from the On-Line Encyclopedia of Integer Sequences (OEIS). It includes the generating function, formulas, examples, and PARI code for computing the sequence.

TYPE: NO_QUESTION

The passage does not contain any explicit mathematical questions or word problems that require solving. It is purely a descriptive entry about a particular integer sequence.