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 A100238 G.f. A(x) satisfies: 2^n + 1 = Sum_{k=0..n} [x^k] A(x)^n for n>=1. 3
 1, 2, -2, 4, -12, 40, -144, 544, -2128, 8544, -35008, 145792, -615296, 2625792, -11311616, 49124352, -214838528, 945350144, -4182412288, 18593224704, -83015133184, 372090122240, -1673660915712, 7552262979584, -34178799378432, 155096251351040, -705533929816064 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Table of n, a(n) for n=0..26. FORMULA a(n) = (-2*(2*n-3)*a(n-1) + 4*(n-3)*a(n-2))/n for n>2, with a(0)=1, a(1)=2, a(2)=-2. G.f.: A(x) = (1+2*x + sqrt(1+4*x-4*x^2))/2. G.f. satisfies: (2+z)^n + (1+z)^n - z^n = Sum_{k=0..n} [x^k] (A(x)+z*x)^n for all z, where [x^k] F(x) denotes the coefficient of x^k in F(x). Given g.f. A(x), then B(x)=A(x)-1-x series reversion is -B(-x). - Michael Somos, Sep 07 2005 Given g.f. A(x) and C(x) = g.f. of A025225, then B(x)=A(x)-1-x satisfies B(x)=x-C(x*B(x)). - Michael Somos, Sep 07 2005 G.f.: 4x^2/(1+2x - sqrt(1+4x-4x^2)). - Michael Somos, Sep 08 2005 EXAMPLE From the table of powers of A(x), we see that 2^n+1 = Sum of coefficients [x^0] through [x^n] in A(x)^n: A^1: [1, 2], -2, 4, -12, 40, -144, 544, -2128, 8544, ...; A^2: [1, 4, 0], 0, -4, 16, -64, 256, -1040, 4288, ...; A^3: [1, 6, 6, -4], 0, 0, -8, 48, -240, 1120, -5088, ...; A^4: [1, 8, 16, 0, -8], 0, 0, 0, -16, 128, -768, ...; A^5: [1, 10, 30, 20, -20, -8], 0, 0, 0, 0, -32, ...; A^6: [1, 12, 48, 64, -12, -48, 0], 0, 0, 0, 0, 0, ...; A^7: [1, 14, 70, 140, 56, -112, -56, 16], 0, 0, 0, ...; A^8: [1, 16, 96, 256, 240, -128, -256, 0, 32], 0, 0, ...; ... In the above table of coefficients in A(x)^n, the main diagonal satisfies: [x^n] A(x)^(n+1) = (n+1)*A009545(n+1) for n>=0. PROG (PARI) {a(n)=if(n==0, 1, (2^n+1-sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j)^n+x*O(x^k), k)))/n)} (PARI) {a(n)=if(n==0, 1, if(n==1, 2, if(n==2, -2, (-2*(2*n-3)*a(n-1)+4*(n-3)*a(n-2))/n)))} (PARI) {a(n)=polcoeff( (1+2*x+sqrt(1+4*x-4*x^2+x^2*O(x^n)))/2, n)} (PARI) a(n)=polcoeff((1+2*x+sqrt(1+4*x-4*x^2+x*O(x^n)))/2, n) CROSSREFS Cf. A100223, A071356, A009545, A143045. a(n) = -(-1)^n * A025227(n), if n>1. Sequence in context: A288303 A287604 A007181 * A208529 A212660 A098774 Adjacent sequences: A100235 A100236 A100237 * A100239 A100240 A100241 KEYWORD sign AUTHOR Paul D. Hanna, Nov 30 2004 STATUS approved

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Last modified November 29 15:54 EST 2023. Contains 367445 sequences. (Running on oeis4.)