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 A291198 G.f. A(x) satisfies: A( 3*x - 2*A(x) ) = x - x^2. 8
 1, 1, -4, 44, -704, 14208, -338304, 9165696, -276371712, 9139825152, -328127311872, 12691848990720, -525849764782080, 23229272756912128, -1089877362984222720, 54133294598753206272, -2838256094009499844608, 156685554517473518682112, -9086394132461874613059584, 552356609566876038974144512, -35128905989147663752065187840, 2333138175889736609287142113280 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS For what fixed integers {a, b, c, d} does there exist a formal power series G(x) with constant term 0 that satisfies G( a*x + b*G(x) ) = c*x + d*x^2 and consists solely of integer coefficients? Necessary condition: there is an integer g_1 with b*g_1^2 + a*g_1 - c = 0, and (a+1)*b*g[1]+a^2+b*c divides d. - Robert Israel, Feb 01 2018 LINKS Paul D. Hanna, Table of n, a(n) for n = 1..300 FORMULA G.f. A(x) satisfies: A( 3*C(x) - 2*A(C(x)) ) = x where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108). a(n) ~ (-1)^n * c * 2^n * n^(log(2)) * n! / (log(2))^n, where c = 0.023113811466383650559... - Vaclav Kotesovec, Aug 28 2017 EXAMPLE G.f.: A(x) = x + x^2 - 4*x^3 + 44*x^4 - 704*x^5 + 14208*x^6 - 338304*x^7 + 9165696*x^8 - 276371712*x^9 + 9139825152*x^10 - 328127311872*x^11 + 12691848990720*x^12 - 525849764782080*x^13 + 23229272756912128*x^14 - 1089877362984222720*x^15 + 54133294598753206272*x^16 +... such that A( 3*x - 2*A(x) ) = x - x^2. RELATED SERIES. Define Ai(x) such that Ai(A(x)) = x, then Ai(x) begins: Ai(x) = x - x^2 + 6*x^3 - 69*x^4 + 1114*x^5 - 22426*x^6 + 530636*x^7 - 14269341*x^8 + 426991250*x^9 - 14018277742*x^10 + 499882371412*x^11 - 19217292112866*x^12 + 791861696758020*x^13 - 34810523545942356*x^14 + 1626245849943319704*x^15 - 80468924895927327693*x^16 +... where Ai(x - x^2) = 3*x - 2*A(x). Thus Ai(x) = 3*C(x) - 2*A(C(x)), where C(x) = x + C(x)^2. MAPLE eq:=A(3*x-2*A(x))+x^2-x: An:= x: for iter from 1 to 6 do R:= map(normal, series(eval(eq, A = unapply(An + add(a[i]*x^i, i=2^(iter-1)+1..2^iter), x)), x, 2^iter+1)); S:= solve({seq(coeff(R, x, i), i=2^(iter-1)..2^iter)}); An:= subs(S, An + add(a[i]*x^i, i=2^(iter-1)+1..2^iter)) od: seq(coeff(An, x, j), j=1..2^6); # Robert Israel, Feb 01 2018 MATHEMATICA nmax = 22; sol = {a[1] -> 1}; Do[A[x_] = Sum[a[k] x^k, {k, 1, n}] /. sol; eq = CoefficientList[A[3x - 2 A[x]] - x + x^2 + O[x]^(n+1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq] [[1]], {n, 1, nmax}]; sol /. Rule -> Set; a /@ Range[1, nmax] (* Jean-François Alcover, Nov 01 2019 *) PROG (PARI) {a(n) = my(A=x, V=[1, 1]); for(i=1, n, V = concat(V, 0); A=x*Ser(V); V[#V] = Vec( subst(A, x, 3*x - 2*A) )[#V] ); V[n]} for(n=1, 30, print1(a(n), ", ")) CROSSREFS Cf. A292812, A292813, A292814, A293454, A293455, A293456. Sequence in context: A056063 A218224 A177749 * A053332 A276369 A157093 Adjacent sequences: A291195 A291196 A291197 * A291199 A291200 A291201 KEYWORD sign AUTHOR Paul D. Hanna, Aug 20 2017 STATUS approved

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Last modified September 24 22:14 EDT 2023. Contains 365582 sequences. (Running on oeis4.)