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A305136
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G.f. A(x) satisfies: 1 = Sum_{n>=0} (A(x)^n - x)^n.
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2
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1, -1, 3, -6, 16, -46, 142, -471, 1606, -5616, 19946, -71659, 260044, -951532, 3507916, -13018488, 48601126, -182406035, 687851096, -2604964819, 9903253875, -37780174088, 144584919872, -554927186056, 2135497844772, -8237988842334, 31850890373750, -123403823544802, 479047178570539, -1863001134095713, 7257404420481641, -28316271490365464, 110645779544592100
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} (A(x)^n - x)^n.
(2) 1 = Sum_{n>=0} A(x)^(n^2) / (1 + x*A(x)^n)^(n+1).
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EXAMPLE
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G.f.: A(x) = x - x^2 + 3*x^3 - 6*x^4 + 16*x^5 - 46*x^6 + 142*x^7 - 471*x^8 + 1606*x^9 - 5616*x^10 + 19946*x^11 - 71659*x^12 + 260044*x^13 - 951532*x^14 + 3507916*x^15 - 13018488*x^16 + 48601126*x^17 - 182406035*x^18 + ...
such that
1 = 1 + (A(x) - x) + (A(x)^2 - x)^2 + (A(x)^3 - x)^3 + (A(x)^4 - x)^4 + (A(x)^5 - x)^5 + (A(x)^6 - x)^6 + (A(x)^7 - x)^7 + ... + (A(x)^n - x)^n + ...
Also,
1 = 1/(1 + x) + A(x)/(1 + x*A(x))^2 + A(x)^4/(1 + x*A(x)^2)^3 + A(x)^9/(1 + x*A(x)^3)^4 + A(x)^16/(1 + x*A(x)^4)^5 + A(x)^25/(1 + x*A(x)^5)^6 + A(x)^36/(1 + x*A(x)^6)^7 + A(x)^49/(1 + x*A(x)^7)^8 + ... + A(x)^(n^2)/(1 + x*A(x)^n)^(n+1) + ...
RELATED SERIES.
Series_Reversion(A(x)) = x + x^2 - x^3 - 4*x^4 - 2*x^5 + 18*x^6 + 40*x^7 - 30*x^8 - 289*x^9 - 346*x^10 + 1151*x^11 + 4319*x^12 + 658*x^13 + ... + A305135(n)*x^n + ...
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PROG
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(PARI) {a(n) = my(A, V=[1]); for(i=1, n, V=concat(V, 0); V[#V] = Vec( sum(m=0, #V+1, (x^m - x*Ser(V))^m ) )[#V+1] ); A = serreverse(x*Ser(V)); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A, V=[1]); for(i=1, n, V=concat(V, 0); V[#V] = Vec( sum(m=0, sqrtint(#V+1), x^(m^2)/(1 + x^m*x*Ser(V))^(m+1) ) )[#V+1] ); A = serreverse(x*Ser(V)); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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