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 A185040 O.g.f.: 1/(1-x) * Sum_{n>=0} 1/(1-(n+1)*x)^n * x^n/n! * exp(-x/(1-(n+1)*x)). 2
 1, 1, 2, 5, 15, 54, 220, 973, 4607, 23230, 124088, 698471, 4124961, 25474314, 164063103, 1099233251, 7645091839, 55085061358, 410472347944, 3158307976315, 25057152530411, 204717532709542, 1720324316575275, 14853374782672785, 131632834029683663, 1196258970969508760 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Compare g.f. to the curious identity: 1/(1+x^2) = Sum_{n>=0} (1-(n+1)*x)^n * x^n/n! * exp(-x*(1-(n+1)*x)). LINKS Table of n, a(n) for n=0..25. FORMULA Binomial transform of A218667. EXAMPLE O.g.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 54*x^5 + 220*x^6 +... where the o.g.f. equals the series: A(x)*(1-x) = exp(-x/(1-x)) + x/(1-2*x)*exp(-x/(1-2*x)) + x^2/(1-3*x)^2/2!*exp(-x/(1-3*x)) + x^3/(1-4*x)^3/3!*exp(-x/(1-4*x)) + x^4/(1-5*x)^4/4!*exp(-x/(1-5*x)) + x^5/(1-6*x)^5/5!*exp(-x/(1-6*x)) + x^6/(1-7*x)^6/6!*exp(-x/(1-7*x)) +... which simplifies to a power series in x with integer coefficients. MATHEMATICA m = 26; Sum[1/(1 - (n+1)x)^n x^n Exp[-x/(1 - (n+1)x)]/n!, {n, 0, m}]/(1-x) + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Jan 27 2020 *) PROG (PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); A=1/(1-x)*sum(k=0, n, 1/(1-(k+1)*X)^k*x^k/k!*exp(-X/(1-(k+1)*X))); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A218667. Sequence in context: A336020 A277175 A056841 * A369599 A208237 A321958 Adjacent sequences: A185037 A185038 A185039 * A185041 A185042 A185043 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 25 2012 STATUS approved

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Last modified August 9 15:15 EDT 2024. Contains 375044 sequences. (Running on oeis4.)