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 A134055 a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(n,k) for n>0, a(0)=1, where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind. 13
 1, 1, 2, 8, 41, 252, 1782, 14121, 123244, 1169832, 11960978, 130742196, 1518514076, 18645970943, 241030821566, 3268214127548, 46338504902485, 685145875623056, 10538790233183702, 168282662416550040, 2784205185437851772, 47646587512911994120 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS FORMULA O.g.f.: Sum_{n>=0} (n*x)^n/(1-n*x)^n * exp(-n*x/(1-n*x)) / n!. - Paul D. Hanna, Nov 04 2012 EXAMPLE O.g.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 41*x^4 + 252*x^5 + 1782*x^6 + 14121*x^7 +... where A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^2*x^2/(1-2*x)^2*exp(-2*x/(1-2*x))/2! + 3^3*x^3/(1-3*x)^3*exp(-3*x/(1-3*x))/3! + 4^4*x^4/(1-4*x)^4*exp(-4*x/(1-4*x))/4! +... simplifies to a power series in x with integer coefficients. Illustrate the definition of the terms by: a(4) = 1*1 + 3*7 + 3*6 + 1*1 = 41; a(5) = 1*1 + 4*15 + 6*25 + 4*10 + 1*1 = 252; a(6) = 1*1 + 5*31 + 10*90 + 10*65 + 5*15 + 1*1 = 1782. MATHEMATICA Flatten[{1, Table[Sum[Binomial[n-1, k-1] * StirlingS2[n, k], {k, 1, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Aug 11 2014 *) PROG (PARI) a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1)*polcoeff(1/prod(i=0, k, 1-i*x +x*O(x^(n-k))), n-k))) (PARI) a(n)=polcoeff(sum(k=0, n+1, (k*x)^k/(1-k*x)^k*exp(-k*x/(1-k*x+x*O(x^n)))/k!), n) for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 04 2012 CROSSREFS Cf. A048993, A122455, A218667, A218670, A245059, A245060. Sequence in context: A067119 A093935 A099240 * A136281 A125698 A231495 Adjacent sequences:  A134052 A134053 A134054 * A134056 A134057 A134058 KEYWORD nonn AUTHOR Paul D. Hanna, Oct 08 2007 EXTENSIONS An initial '1' was added and definition changed slightly by Paul D. Hanna, Nov 04 2012 STATUS approved

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Last modified September 22 20:30 EDT 2018. Contains 315270 sequences. (Running on oeis4.)