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A203067
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G.f. satisfies: A(x) = Product_{n>=0} 1/( (1 - (x*A(x))^(5*n+1)) * (1 - (x*A(x))^(5*n+4)) ).
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1
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1, 1, 2, 5, 15, 48, 161, 557, 1975, 7140, 26223, 97567, 366979, 1393104, 5330500, 20537289, 79605148, 310211668, 1214618931, 4776089725, 18852665026, 74676503735, 296736116911, 1182534483734, 4725095364115, 18926460148937, 75981766859715, 305674059089303, 1232118979522167
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OFFSET
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0,3
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COMMENTS
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G.f. is an eigenfunction of the Rogers-Ramanujan identity described by A003114: Sum_{n>=0} x^(n^2)/(Product_{k=1..n} 1-x^k) = Product_{n>=0} 1/((1-x^(5*n+1))*(1-x^(5*n+4))).
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LINKS
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FORMULA
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G.f. satisfies:
(1) A(x) = Sum_{n>=0} x^(n^2)*A(x)^(n^2) / (Product_{k=1..n} 1 - x^k*A(x)^k).
(2) A(x) = (Product_{n>0} 1 + (x*A(x))^(2*n)) * (Sum_{n>=0} (x*A(x))^(n^2) / (Product_{k=1..n} 1 - (x*A(x))^(4*k))).
(3) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A003114.
(4) A(x) = (1/x)*Series_Reversion(x/G(x)) where G(x) is the g.f. of A003114.
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 48*x^5 + 161*x^6 + 557*x^7 +...
where the g.f. satisfies the Rogers-Ramanujan identity:
A(x) = 1/( (1 - x*A(x))*(1 - x^4*A(x)^4) * (1 - x^6*A(x)^6)*(1 - x^9*A(x)^9) * (1 - x^11*A(x)^11)*(1 - x^14*A(x)^14) * (1 - x^16*A(x)^16)*(1 - x^19*A(x)^19) *...);
A(x) = 1 + x*A(x)/(1-x*A(x)) + x^4*A(x)^4/((1-x*A(x))*(1-x^2*A(x)^2)) + x^9*A(x)^9/((1-x*A(x))*(1-x^2*A(x)^2)*(1-x^3*A(x)^3)) +...
Also, A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A003114:
G(x) = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + 6*x^10 + 7*x^11 + 9*x^12 +...
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PROG
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(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), (x*A)^(m^2)/prod(k=1, m, 1-(x*A)^k+x*O(x^n)))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=prod(m=0, n, 1/((1-(x*A)^(5*m+1))*(1-(x*A)^(5*m+4)))+x*O(x^n))); polcoeff(A, n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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