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 A198865 E.g.f. satisfies: A(x) = 1 + sinh(x*A(x)^2). 0
 1, 1, 4, 31, 368, 5941, 121632, 3019563, 88140544, 2958267241, 112246484480, 4751313955543, 221980968007680, 11346405913579101, 629859586327810048, 37736053514310470371, 2426956220333852131328, 166775317658298155269585, 12195158366650225121427456 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Table of n, a(n) for n=0..18. FORMULA (1) E.g.f. satisfies: A( x/(1 + sinh(x))^2 ) = 1 + sinh(x). (2) E.g.f.: A(x) = sqrt( Series_Reversion( x/(1 + sinh(x))^2 ) / x ). (3) a(n) = [x^n/n!] (1 + sinh(x))^(2*n+1) / (2*n+1). (4) a(n) = Sum_{k=0..n} C(2*n+1,k)/(2*n+1)*Sum_{j=0..k} (-1)^(k-j)*C(k,j)*(2j-k)^n/2^k. (5) Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then a(n,m) = Sum_{k=0..n} m*C(2*n+m,k)/(2*n+m)*Sum_{j=0..k} (-1)^(k-j)*C(k,j)*(2j-k)^n/2^k. a(n) ~ s*sqrt((2-2*s+s^2)/(2*(2-3*s+2*s^2))) * n^(n-1) * (2*s*sqrt(2-2*s+s^2))^n / exp(n), where s = 1.75931315231552523... is the root of the equation 2*sqrt(2+(s-2)*s) * log(1+sqrt(1+(1-s)^2)-s) = -s. - Vaclav Kotesovec, Jan 11 2014 EXAMPLE E.g.f.: A(x) = 1 + x + 4*x^2/2! + 31*x^3/3! + 368*x^4/4! + 5941*x^5/5! +... Related expansions. A(x)^2 = 1 + 2*x + 10*x^2/2! + 86*x^3/3! + 1080*x^4/4! + 18042*x^5/5! +... 1/(1 + sinh(x))^2 = 1 - 2*x + 6*x^2/2! - 26*x^3/3! + 144*x^4/4! - 962*x^5/5! +... Coefficients of [x^n/n!] in the odd powers of (1 + sinh(x)) begin: 1: [(1), 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,...]; 3: [1,(3), 6, 9, 24, 63, 96, 549, 384, 4923, 1536, 44289,...]; 5: [1, 5,(20), 65, 200, 725, 2720, 9665, 41600, 165125,...]; 7: [1, 7, 42,(217), 1008, 4627, 22512, 112357, 567168,...]; 9: [1, 9, 72, 513,(3312), 20169, 122112, 756513, 4770432,...]; 11:[1, 11, 110, 1001, 8360,(65351), 492800, 3693701, 27948800,...]; 13:[1, 13, 156, 1729, 17784, 171613,(1581216), 14210209,...]; 15:[1, 15, 210, 2745, 33600, 387675, 4262160,(45293445),...]; 17:[1, 17, 272, 4097, 58208, 783377, 10057472, 124378817,(1498389248), ...]; ... where the coefficients in parenthesis generate this sequence like so: [1, 3/3, 20/5, 217/7, 3312/9, 65351/11, 1581216/13, 45293445/15,...]. MATHEMATICA CoefficientList[Sqrt[InverseSeries[Series[x/(1 + Sinh[x])^2, {x, 0, 21}], x]/x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 11 2014 *) PROG (PARI) /* PARI programs for a(n, m) where A(x)^m = Sum_{n>=0} a(n, m)*x^n/n! */ {a(n, m=1)=n!*polcoeff((1+sinh(x +x*O(x^n)))^(2*n+m)*m/(2*n+m), n)} (PARI) {a(n, m=1)=sum(k=0, n, m*binomial(2*n+m, k)/(2*n+m)*sum(j=0, k, (-1)^(k-j)*binomial(k, j)*(2*j-k)^n/2^k))} (PARI) {a(n, m=1)=n!*polcoeff((serreverse(x/(1+sinh(x +x*O(x^n)))^2)/x)^(m/2), n)} CROSSREFS Cf. A162653. Sequence in context: A138860 A266757 A369737 * A145087 A215529 A351798 Adjacent sequences: A198862 A198863 A198864 * A198866 A198867 A198868 KEYWORD nonn AUTHOR Paul D. Hanna, Oct 30 2011 STATUS approved

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