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 A368593 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*sqrt( (A(x)^4 + A(-x)^4)/2 ). 3
 1, 1, 2, 7, 18, 78, 220, 1043, 3090, 15402, 47044, 242126, 755076, 3973820, 12580344, 67303139, 215511330, 1167556434, 3772175860, 20640707866, 67167649868, 370510806212, 1212836703304, 6735128062542, 22156120392276, 123731147310820, 408741630687656, 2293595176625340 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0, and even elsewhere. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..600 FORMULA G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas. (1) A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*sqrt( (A(x)^4 + A(-x)^4)/2 ). (2) A(x) = 2 - A(-x) + x*A(x)^2 - x*A(-x)^2. (3) A(x) = A(-x) + x*sqrt( (A(x)^4 + x*A(-x)^4)/2 ). (4.a) A(x) = (1 - sqrt(1-8*x + 4*x*A(-x) + 4*x^2*A(-x)^2)) / (2*x). (4.b) A(-x) = (sqrt(1+8*x - 4*x*A(x) + 4*x^2*A(x)^2) - 1) / (2*x). (5) (A(x) + A(-x))/2 = 1/(1 - x*(A(x) - A(-x))). EXAMPLE G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 18*x^4 + 78*x^5 + 220*x^6 + 1043*x^7 + 3090*x^8 + 15402*x^9 + 47044*x^10 + 242126*x^11 + 755076*x^12 + ... RELATED SERIES. A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 54*x^4 + 220*x^5 + 717*x^6 + 3090*x^7 + 10562*x^8 + 47044*x^9 + 165858*x^10 + 755076*x^11 + ... A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 205*x^4 + 836*x^5 + 3178*x^6 + 13192*x^7 + 51490*x^8 + 216808*x^9 + 862588*x^10 + ... The even bisection of A(x) may be formed from the odd bisection of A(x)^2: (A(x) + A(-x))/2 = 1 + 2*x^2 + 18*x^4 + 220*x^6 + 3090*x^8 + 47044*x^10 + ... (A(x)^2 - A(-x)^2)/2 = 2*x + 18*x^3 + 220*x^5 + 3090*x^7 + 47044*x^9 + ... The odd bisection of A(x) may be formed from the even bisection of A(x)^4: (A(x) - A(-x))/2 = x + 7*x^3 + 78*x^5 + 1043*x^7 + 15402*x^9 + ... (A(x)^4 + A(-x)^4)/2 = 1 + 14*x^2 + 205*x^4 + 3178*x^6 + 51490*x^8 + ... sqrt( (A(x)^4 + A(-x)^4)/2 ) = 1 + 7*x^2 + 78*x^4 + 1043*x^6 + 15402*x^8 + ... PROG (PARI) {a(n) = my(A=[1]); for(i=1, n, A = Vec(1 + x*(Ser(A)^2 - subst(Ser(A)^2, x, -x))/2 + x*sqrt( (Ser(A)^4 + subst(Ser(A)^4, x, -x))/2 ) +x*O(x^#A) ) ); A[n+1]} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A368628, A368629, A368626, A368627. Sequence in context: A002214 A303742 A340738 * A218684 A337614 A343908 Adjacent sequences: A368590 A368591 A368592 * A368594 A368595 A368596 KEYWORD nonn AUTHOR Paul D. Hanna, Jan 10 2024 STATUS approved

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Last modified August 5 15:49 EDT 2024. Contains 374950 sequences. (Running on oeis4.)