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 A318008 G.f. A(x) satisfies: A( A( x - x^2 ) ) = x + x^2. 3
 1, 1, 1, 2, 4, 9, 21, 50, 122, 302, 758, 1928, 4958, 12849, 33509, 88122, 233810, 621022, 1641150, 4411180, 12364368, 33073210, 71807506, 206985492, 1354944972, 3153779248, -33794258600, -62697691948, 2524565441138, 5004344042337, -186642439700891, -368380986364150, 16196862324254354, 32039943659306982, -1602823227559245434 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS a(2*n-1) = A277292(n). a(2^k-1) = 1 (mod 2) and a(2^(k+1)-2) = 1 (mod 2) for k >= 1, and a(n) is even elsewhere (conjecture). LINKS Paul D. Hanna, Table of n, a(n) for n = 1..300 FORMULA G.f. A(x) satisfies: (1) A(-A(-x)) = x. (2a) A(A(x)) = 2*C(x) - x, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108). (2b) A(A( x - x^2 )) = x + x^2. (2c) A(A( x/(1+x)^2 )) = (x + 2*x^2)/(1+x)^2. (3a) (A(x) - A(-x))^2 = 2*(A(x) + A(-x)). (3b) A(x)^2 - 2*A(x) - 2*A(x)*A(-x) + A(-x)^2 - 2*A(-x) = 0. Define B(x) = (A(x) - A(-x))/2 and Catalan series C(x) = x + C(x)^2, then (4a) B(x)^2 = (A(x) + A(-x))/2. (4b) A(x) = B(x) + B(x)^2. (5a) B( A(x - x^2) ) = x. (5b) B( A(x) ) = C(x). (6a) A( B(x) - B(x)^2 ) = x. (6b) B( B(x) + B(x)^2 ) = C(x). (6c) C( B(x) - B(x)^2 ) = B(x). EXAMPLE G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 9*x^6 + 21*x^7 + 50*x^8 + 122*x^9 + 302*x^10 + 758*x^11 + 1928*x^12 + 4958*x^13 + 12849*x^14 + 33509*x^15 + ... such that A(A(x - x^2)) = x + x^2. RELATED SERIES. (a) A(A(x)) = x + 2*x^2 + 4*x^3 + 10*x^4 + 28*x^5 + 84*x^6 + 264*x^7 + 858*x^8 + 2860*x^9 + 9724*x^10 + ... + 2*A000108(n-2)*x^n + ... (b) The odd bisection B(x) = (A(x) - A(-x))/2 begins B(x) = x + x^3 + 4*x^5 + 21*x^7 + 122*x^9 + 758*x^11 + 4958*x^13 + 33509*x^15 + 233810*x^17 + 1641150*x^19 + 12364368*x^21 + ... + A277292(n)*x^(2*n-1) + ... such that B(x)^2 yields the even bisection (A(x) + A(-x))/2 B(x)^2 = x^2 + 2*x^4 + 9*x^6 + 50*x^8 + 302*x^10 + 1928*x^12 + 12849*x^14 + 88122*x^16 + 621022*x^18 + 4411180*x^20 + ... + a(2*n)*x^(2*n) + ... thus A(x) = B(x) + B(x)^2. (c) Also, the Catalan series equals B( B(x) + B(x)^2 ) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + ... + A000108(n-2)*x^n + ... (d) Note that A(x - x^2) equals the series reversion of B(x): A(x - x^2) = x - x^3 - x^5 - x^7 + 3*x^9 + 11*x^11 - 9*x^13 + 71*x^15 - 1685*x^17 + 31683*x^19 - 845729*x^21 + 28968319*x^23 + ... where B( A(x -x^2) ) = x and A( B(x) - B(x)^2 ) = x. PROG (PARI) /* Using A(A( x - x^2 )) = x + x^2. */ {a(n) = my(A=x+x*O(x^n)); for(i=1, n, A = A + (x+x^2 - subst(A, x, subst(A, x, x-x^2)) )/2 ); polcoeff(A, n)} for(n=1, 30, print1(a(n), ", ")) CROSSREFS Cf. A277292, A000108. Sequence in context: A092423 A238438 A257104 * A199410 A091600 A261232 Adjacent sequences: A318005 A318006 A318007 * A318009 A318010 A318011 KEYWORD sign AUTHOR Paul D. Hanna, Sep 06 2018 STATUS approved

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Last modified February 26 07:39 EST 2024. Contains 370335 sequences. (Running on oeis4.)