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A240956
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G.f.: Sum_{n>=0} n^n * x^n * (2 + n*x)^n / (1 + n*x)^(2*n+2).
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4
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1, 2, 9, 72, 830, 12420, 228060, 4959360, 124589304, 3550050000, 113116311000, 3985174226880, 153815533185600, 6454433029524480, 292557975636326400, 14244829479956275200, 741502151945703308160, 41092028680670274827520, 2415394879269218890243200
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OFFSET
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0,2
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LINKS
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FORMULA
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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
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EXAMPLE
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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 +...
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MATHEMATICA
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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 *)
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PROG
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(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), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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