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Search: id:A134774
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| A134774 |
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G.f.: A(x) = Product_{n>=1} G(x^n,n)^n where G(x,n) = 1 + x*G(x,n)^n. |
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+0
2
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| 1, 1, 3, 6, 15, 26, 66, 110, 253, 460, 966, 1680, 3732, 6304, 13073, 23539, 47548, 82362, 171463, 293578, 597934, 1056830, 2105424, 3654919, 7533609, 12915780, 26112978, 46033557, 92504870, 160298673, 330468463, 568239653, 1161488784
(list; graph; listen)
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OFFSET
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0,3
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FORMULA
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G.f.: A(x) = exp( Sum_{n>=1} A105862(n)/n*x^n ), where A105862(n) = Sum_{d|n} binomial(n,d)*n/GCD(n,d). G.f.: A(x) = Product_{n>=1} [ Series_Reversion( x/(1 + x^n) )/x ]^n.
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 26*x^5 + 66*x^6 +...
G.f.: A(x) = 1/(1-x) * G(x^2,2)^2 * G(x^3,3)^3 * G(x^4,4)^4 *...
where the functions G(x,n) are g.f.s of well-known sequences:
G(x,2) = g.f. of A000108 = 1 + x*G(x,2)^2;
G(x,3) = g.f. of A001764 = 1 + x*G(x,3)^3;
G(x,4) = g.f. of A002293 = 1 + x*G(x,4)^4 ; etc.
Explicitly, the product yielding the g.f. A(x) begins:
A(x) = [1 + x + x^2 + x^3 +...] * [1 + 2*x^2 + 5*x^4 + 14*x^6 +...] * [1 + 3*x^3 + 12*x^6 + 55*x^9 +...] * [1 + 4*x^4 + 22*x^8 + 140*x^12 +...] * ...
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PROGRAM
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(PARI) {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, x^m*sumdiv(m, d, binomial(m, d)/gcd(m, d)))), n))} (PARI) {a(n)=polcoeff(prod(m=1, n, (1/x*serreverse(x/(1+x^m +x*O(x^n))))^m), n)}
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CROSSREFS
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Cf. A105862 (LOG(A(x)); A056045 (variant); A000108 (Catalan), A001764, A002293.
Adjacent sequences: A134771 A134772 A134773 this_sequence A134775 A134776 A134777
Sequence in context: A062878 A020991 A079825 this_sequence A056278 A000740 A069712
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Nov 11 2007
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