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 A193287 E.g.f.: A(x) = 1/(1 - 2*x^2)^(1 + 1/(2*x)). 4
 1, 1, 5, 19, 145, 981, 10141, 98575, 1289569, 16314121, 258568021, 4023553931, 74961787825, 1383475135069, 29636315118957, 632414472704071, 15316605861040321, 370875832116841105, 10021723060544059429, 271409166367070755843 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS More generally, we have the identity: Sum_{n>=0} (x^n/n!)*Product_{k=1..n} (1+k*y) = 1/(1 - x*y)^(1 + 1/y); here y=2*x. LINKS FORMULA E.g.f.: A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} (1 + 2*k*x). a(n) ~ n! * 2^(n/2-1/2-1/sqrt(2))*n^(1/sqrt(2))/Gamma(1/sqrt(2)). - Vaclav Kotesovec, Jun 25 2013 EXAMPLE E.g.f.: A(x) = 1 + x + 5*x^2/2! + 19*x^3/3! + 145*x^4/4! + 981*x^5/5! +... where A(x) satisfies: A(x)^(2*x/(1+2*x)) = 1 + 2*x^2 + 4*x^4 + 8*x^6 + 16*x^8 + 32*x^10 +... Also, A(x) = 1 + x*(1+2*x) + x^2*(1+2*x)*(1+4*x)/2! + x^3*(1+2*x)*(1+4*x)*(1+6*x)/3! + x^4*(1+2*x)*(1+4*x)*(1+6*x)*(1+8*x)/4! +... The logarithm begins: log(A(x)) = x + 2*x^2 + 2*x^3/2 + 4*x^4/2 + 4*x^5/3 + 8*x^6/3 + 8*x^7/4 +... MATHEMATICA CoefficientList[Series[1/(1-2*x^2)^(1+1/(2*x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 25 2013 *) PROG (PARI) {a(n)=n!*polcoeff(1/(1 - 2*x^2 +x^2*O(x^n))^((1+2*x)/(2*x)), n)} (PARI) {a(n)=n!*polcoeff(sum(m=0, n, x^m/m!*prod(k=1, m, 1+2*k*x+x*O(x^n))), n)} CROSSREFS Cf. A193281, A193288, A193289, A193290. Sequence in context: A297389 A228479 A187018 * A027269 A082790 A145935 Adjacent sequences:  A193284 A193285 A193286 * A193288 A193289 A193290 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 21 2011 STATUS approved

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Last modified June 17 19:57 EDT 2021. Contains 345085 sequences. (Running on oeis4.)