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A276363
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G.f. satisfies: A(x - 3*A(x)^2) = x + A(x)^2.
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19
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1, 4, 56, 1172, 30248, 892296, 28951344, 1010322900, 37384819496, 1452697058744, 58872642043856, 2475764515398568, 107619880380347920, 4821324372637921744, 222077355203506939104, 10497354682052048593332, 508414637258604924680136, 25197644191294099697736312, 1276547957544912412461457680, 66046883289153773427379134360, 3487101507192780951408327918192
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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) also satisfies:
(1) A(x) = x + 4 * A( x/4 + 3*A(x)/4 )^2.
(2) 3*A(x) = -x + 4 * Series_Reversion(x - 3*A(x)^2).
(3) R(x) = -3*x + 4 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/4 - R(x)/4 ) ) = 3*x/4 + R(x)/4, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k)*3^k*4^(n-k-1).
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EXAMPLE
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G.f.: A(x) = x + 4*x^2 + 56*x^3 + 1172*x^4 + 30248*x^5 + 892296*x^6 + 28951344*x^7 + 1010322900*x^8 + 37384819496*x^9 + 1452697058744*x^10 + 58872642043856*x^11 +
2475764515398568*x^12 +...
such that A(x - 3*A(x)^2) = x + A(x)^2.
RELATED SERIES.
Note that Series_Reversion(x - 3*A(x)^2) = x/4 + 3*A(x)/4, which begins:
Series_Reversion(x - 3*A(x)^2) = x + 3*x^2 + 42*x^3 + 879*x^4 + 22686*x^5 + 669222*x^6 + 21713508*x^7 + 757742175*x^8 + 28038614622*x^9 + 1089522794058*x^10 + 44154481532892*x^11 + 1856823386548926*x^12 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 4*x^2 - 24*x^3 - 372*x^4 - 7944*x^5 - 204168*x^6 - 5942256*x^7 - 189500916*x^8 - 6490281480*x^9 - 235609789368*x^10 - 8983294304784*x^11 - 357373688297448*x^12 +...
then Series_Reversion(x + A(x)^2) = 3*x/4 + R(x)/4.
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MATHEMATICA
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m = 22; A[_] = 0;
Do[A[x_] = x + 4A[x/4 + 3A[x]/4]^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 - 3*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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