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 A240956 G.f.: Sum_{n>=0} n^n * x^n * (2 + n*x)^n / (1 + n*x)^(2*n+2). 4
 1, 2, 9, 72, 830, 12420, 228060, 4959360, 124589304, 3550050000, 113116311000, 3985174226880, 153815533185600, 6454433029524480, 292557975636326400, 14244829479956275200, 741502151945703308160, 41092028680670274827520, 2415394879269218890243200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 FORMULA a(n) = Sum_{k=0..[n/2]} (n-k)! * Stirling2(n, n-k) * binomial(n-k, k) * 2^(n-2*k). a(n) ~ c * d^n * n! / sqrt(n), where d = 2*r^2/(2*r-1) + (2*r-1)*r/(2*(1-r)) = 3.36074272900128370245729732045120604190737486342012..., where r = 0.80276231206743119172295651485200150958072822575039811732... is the root of the equation (r + (1-2*r)^2/(4*(1-r))) * LambertW(-exp(-1/r)/r) = -1, and c = 0.533888836381702228067397487907088688592161798613354080016... . - Vaclav Kotesovec, Aug 05 2014 EXAMPLE O.g.f.: A(x) = 1 + 2*x + 9*x^2 + 72*x^3 + 830*x^4 + 12420*x^5 + 228060*x^6 +... where A(x) = 1 + x*(2+x)/(1+x)^4 + 2^2*x^2*(2+2*x)^2/(1+2*x)^6 + 3^3*x^3*(2+3*x)^3/(1+3*x)^8 + 4^4*x^4*(2+4*x)^4/(1+4*x)^10 + 5^5*x^5*(2+5*x)^5/(1+5*x)^12 +... MATHEMATICA Table[Sum[(n-k)! * StirlingS2[n, n-k] * Binomial[n-k, k] * 2^(n-2*k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 05 2014 *) PROG (PARI) {a(n)=local(A=1); A=sum(m=0, n, m^m*x^m*(2+m*x)^m/(1 + m*x +x*O(x^n))^(2*m+2)); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) (PARI) /* From formula for a(n): */ {Stirling2(n, k)=sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!} {a(n)=sum(k=0, n\2, (n-k)!*Stirling2(n, n-k)*binomial(n-k, k)*2^(n-2*k) )} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A240957, A240958. Sequence in context: A118789 A258114 A133941 * A038035 A133984 A208898 Adjacent sequences:  A240953 A240954 A240955 * A240957 A240958 A240959 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 04 2014 STATUS approved

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Last modified May 19 22:32 EDT 2019. Contains 323411 sequences. (Running on oeis4.)