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A307955
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G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * ((1+x)^(5*n) - A(x))^(n+1), where A(0) = 0.
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4
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1, 9, 46, 344, 3586, 39676, 490036, 6669184, 97419116, 1519635734, 25170406452, 439941245653, 8081132624095, 155483518553143, 3124130586316551, 65389133324807724, 1422540686845941509, 32103883123046977644, 750278496443387818395, 18128963984900687497993, 452255024819251695443556, 11632687351726270908152086, 308130679955484625602559961, 8395760218678197725930082459
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} x^n * ((1+x)^(5*n) - A(x))^(n+1).
(2) 1 + x = Sum_{n>=0} x^n * (1+x)^(5*n*(n-1)) / (1 + x*(1+x)^(5*n)*A(x))^(n+1).
(3) 1 = Sum_{n>=0} x^n * (1-x)^(10*n+2) / ((1-x)^(5*n+1) - x*A(x/(1-x)))^(n+1).
(4) 1 = Sum_{n>=0} x^n * (1 - (1-x)^(5*n-5) * A(x/(1-x)))^n / (1-x)^(5*n^2-4*n-1)).
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EXAMPLE
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G.f.: A(x) = x + 9*x^2 + 46*x^3 + 344*x^4 + 3586*x^5 + 39676*x^6 + 490036*x^7 + 6669184*x^8 + 97419116*x^9 + 1519635734*x^10 + 25170406452*x^11 + ...
such that
1 = (1 - A(x)) + x*((1+x)^5 - A(x))^2 + x^2*((1+x)^10 - A(x))^3 + x^3*((1+x)^15 - A(x))^4 + x^4*((1+x)^20 - A(x))^5 + x^5*((1+x)^25 - A(x))^6 + x^6*((1+x)^30 - A(x))^7 + x^7*((1+x)^35 - A(x))^8 + ...
also
1 + x = 1/(1 + x*A(x)) + x/(1 + x*(1+x)^5*A(x))^2 + x^2*(1+x)^10/(1 + x*(1+x)^10*A(x))^3 + x^3*(1+x)^30/(1 + x*(1+x)^15*A(x))^4 + x^4*(1+x)^60/(1 + x*(1+x)^20*A(x))^5 + x^5*(1+x)^100/(1 + x*(1+x)^25*A(x))^6 + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n-1, A = concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, x^m*((1+x +x*O(x^#A))^(5*m) - x*Ser(A))^(m+1) ), #A); ); A[n]}
for(n=1, 30, print1(a(n), ", ")) \\ shifted by Georg Fischer, Jun 22 2022
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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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