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A307954
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G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * ((1+x)^(4*n) - A(x))^(n+1), where A(0) = 0.
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4
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1, 7, 29, 199, 1799, 17227, 186329, 2206267, 28014806, 379929320, 5463954442, 82846605311, 1319217639773, 21986394800481, 382391651777315, 6923247705041891, 130203668578601251, 2538723618787364104, 51232176790185969490, 1068400287907868926763, 22992012848929364898826, 509930616374506270683390, 11641797557959754404475921, 273291680187022711826902024
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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)^(4*n) - A(x))^(n+1).
(2) 1 + x = Sum_{n>=0} x^n * (1+x)^(4*n*(n-1)) / (1 + x*(1+x)^(4*n)*A(x))^(n+1).
(3) 1 = Sum_{n>=0} x^n * (1-x)^(8*n+2) / ((1-x)^(4*n+1) - x*A(x/(1-x)))^(n+1).
(4) 1 = Sum_{n>=0} x^n * (1 - (1-x)^(4*n-4) * A(x/(1-x)))^n / (1-x)^(4*n^2-3*n-1)).
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EXAMPLE
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G.f.: A(x) = x + 7*x^2 + 29*x^3 + 199*x^4 + 1799*x^5 + 17227*x^6 + 186329*x^7 + 2206267*x^8 + 28014806*x^9 + 379929320*x^10 + 5463954442*x^11 + ...
such that
1 = (1 - A(x)) + x*((1+x)^4 - A(x))^2 + x^2*((1+x)^8 - A(x))^3 + x^3*((1+x)^12 - A(x))^4 + x^4*((1+x)^16 - A(x))^5 + x^5*((1+x)^20 - A(x))^6 + x^6*((1+x)^24 - A(x))^7 + x^7*((1+x)^28 - A(x))^8 + ...
also
1 + x = 1/(1 + x*A(x)) + x/(1 + x*(1+x)^4*A(x))^2 + x^2*(1+x)^8/(1 + x*(1+x)^8*A(x))^3 + x^3*(1+x)^24/(1 + x*(1+x)^12*A(x))^4 + x^4*(1+x)^48/(1 + x*(1+x)^16*A(x))^5 + x^5*(1+x)^80/(1 + x*(1+x)^20*A(x))^6 + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, x^m*((1+x +x*O(x^#A))^(4*m) - x*Ser(A))^(m+1) ), #A); ); A[n+1]}
for(n=0, 30, print1(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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