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A276365
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G.f. A(x) satisfies: A(x - 2*A(x)^2) = x - A(x)^2.
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3
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1, 1, 6, 53, 578, 7234, 100044, 1495125, 23802346, 399740086, 7032766196, 128952474242, 2454645604820, 48359400068836, 983683769369624, 20618782389897333, 444636132851851386, 9851377271964349038, 223998085060636514020, 5221799494107885481430, 124695762315403816775932, 3047952254964607540099676, 76206565881709345978097960, 1947752912315470845518308642, 50860833685759573411702643972
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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) also satisfies:
(1) A(x) = x + A( 2*A(x) - x )^2.
(2) 2*A(x) = x + Series_Reversion(x - 2*A(x)^2).
(3) R(x) = 2*x - Series_Reversion(x - A(x)^2), where R(A(x)) = x.
(4) R( (x - R(x))^(1/2) ) = 2*x - R(x), where R(A(x)) = x.
(5) A(x) = x + Sum_{n>=1} 2^(n-1) * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
(6) A(x) = x + G(A(x))^2, where G(x) = sqrt(x - R(x)) is the g.f. of A177409, and R(A(x)) = x. - Paul D. Hanna, Nov 18 2022
a(n) = Sum_{k=0..n-1} A277295(n,k)*2^k.
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EXAMPLE
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G.f.: A(x) = x + x^2 + 6*x^3 + 53*x^4 + 578*x^5 + 7234*x^6 + 100044*x^7 + 1495125*x^8 + 23802346*x^9 + 399740086*x^10 + 7032766196*x^11 +...
such that A(x - 2*A(x)^2) = x - A(x)^2.
RELATED SERIES.
Note that Series_Reversion(x - 2*A(x)^2) = 2*A(x) - x, which begins:
Series_Reversion(x - 2*A(x)^2) = x + 2*x^2 + 12*x^3 + 106*x^4 + 1156*x^5 + 14468*x^6 + 200088*x^7 + 2990250*x^8 + 47604692*x^9 + 799480172*x^10 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - x^2 - 4*x^3 - 28*x^4 - 264*x^5 - 2992*x^6 - 38496*x^7 - 544464*x^8 - 8298080*x^9 - 134500672*x^10 - 2297361024*x^11 +...
then Series_Reversion(x - A(x)^2) = 2*x - R(x), and
R(x) = x - G(x)^2, where G(x) = x + 2*x^2 + 12*x^3 + 108*x^4 + 1208*x^5 + 15536*x^6 + 220832*x^7 + 3390480*x^8 + ... + A177409(n)*x^n + ...
Also, sqrt(A(x) - x) = A(2*A(x) - x), which begins:
sqrt(A(x) - x) = x + 3*x^2 + 22*x^3 + 223*x^4 + 2706*x^5 + 36998*x^6 + 552172*x^7 + 8827263*x^8 + 149328698*x^9 + 2650946274*x^10 + ...
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MATHEMATICA
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m = 26; A[_] = 0;
Do[A[x_] = x + A[2 A[x] - x]^2 + O[x]^m // Normal, {m}];
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PROG
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(PARI) {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - 2*F^2) + F^2, #A) ); A[n]}
for(n=1, 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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