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A145268
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G.f. A(x) satisfies A(x) = 1/Product_{k>0} (1-x^k*A(x)).
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8
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1, 1, 3, 9, 30, 104, 378, 1414, 5424, 21208, 84244, 339008, 1379173, 5663078, 23439651, 97692524, 409650348, 1727034770, 7315915371, 31124324364, 132926220818, 569695276362, 2449395461726, 10561857055472, 45664873651576
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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FORMULA
| G.f. A(x) satisfies:
(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^n / Product_{k=1..n} (1-x^k) due to an identity of Euler. [From Paul D. Hanna, May 21 2011]
(2) A(x) = 1 + Sum_{n>=1} x^(n^2)*A(x)^n / [Product_{k=1..n} (1-x^k)*(1-x^k*A(x))] due to Cauchy's identity. [From Paul D. Hanna, May 21 2011]
(3) A(x) = 1 + Sum_{n>=1} x^n*A(x) / Product_{k=1..n} (1 - x^k*A(x)) due to an identity of Euler. [From Paul D. Hanna, Feb 11 2012]
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EXAMPLE
| G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 30*x^4 + 104*x^5 + 378*x^6 +...
The g.f. satisfies:
(0) A(x) = 1/((1-x*A(x)) * (1-x^2*A(x)) * (1-x^3*A(x)) *...).
(1) A(x) = 1 + x*A(x)/(1-x) + x^2*A(x)^2/((1-x)*(1-x^2)) + x^3*A(x)^3/((1-x)*(1-x^2)*(1-x^3)) +...
(2) A(x) = 1 + x*A(x)/[(1-x)*(1-x*A(x))] + x^4*A(x)^2/[(1-x)*(1-x^2)*(1-x*A(x))*(1-x^2*A(x))] + x^9*A(x)^3/[(1-x)*(1-x^2)*(1-x^3)*(1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))] +...
(3) A(x) = 1 + x*A(x)/(1-x*A(x)) + x^2*A(x)/((1-x*A(x))*(1-x^2*A(x))) + x^3*A(x)/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...
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PROG
| (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-x^k*A+x*O(x^n)))); polcoeff(A, n)} /* From Paul D. Hanna */
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^m/prod(k=1, m, (1-x^k+x*O(x^n))))); polcoeff(A, n)} /* From Paul D. Hanna */
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*A^m/prod(k=1, m, (1-x^k)*(1-x^k*A+x*O(x^n))))); polcoeff(A, n)} /* From Paul D. Hanna */
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A/prod(k=1, m, (1-x^k*A+x*O(x^n))))); polcoeff(A, n)} /* From Paul D. Hanna */
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CROSSREFS
| Cf. A145267, A196150.
Sequence in context: A148954 A148955 A119372 * A148956 A003409 A029651
Adjacent sequences: A145265 A145266 A145267 * A145269 A145270 A145271
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KEYWORD
| easy,nonn,changed
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AUTHOR
| Vladeta Jovovic (vladeta(AT)eunet.yu), Oct 05 2008
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EXTENSIONS
| More terms from Max Alekseyev (maxale(AT)gmail.com), Jan 31 2010
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