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A206154
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a(n) = Sum_{k=0..n} binomial(n,k)^(k+2).
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4
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1, 2, 10, 110, 2386, 125752, 14921404, 3697835668, 2223231412546, 3088517564289836, 9040739066816429380, 63462297965044771663708, 1064766030857977088480630740, 37863276208844960432962611293828, 3144384748384240804260912067907833280
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OFFSET
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0,2
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COMMENTS
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Ignoring initial term a(0), equals the logarithmic derivative of A206153.
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LINKS
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Table of n, a(n) for n=0..14.
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FORMULA
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Limit n->infinity a(n)^(1/n^2) = (1-r)^(-r/2) = 1.53362806511..., where r = 0.70350607643... (see A220359) is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Jan 29 2014
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EXAMPLE
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L.g.f.: L(x) = 2*x + 10*x^2/2 + 110*x^3/3 + 2386*x^4/4 + 125752*x^5/5 +...
where exponentiation yields A206151:
exp(L(x)) = 1 + 2*x + 7*x^2 + 48*x^3 + 693*x^4 + 26632*x^5 + 2542514*x^6 +...
Illustration of initial terms:
a(1) = 1^2 + 1^3 = 2;
a(2) = 1^2 + 2^3 + 1^4 = 10;
a(3) = 1^2 + 3^3 + 3^4 + 1^5 = 110;
a(4) = 1^2 + 4^3 + 6^4 + 4^5 + 1^6 = 2386;
a(5) = 1^2 + 5^3 + 10^4 + 10^5 + 5^6 + 1^7 = 125752; ...
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MATHEMATICA
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Table[Sum[Binomial[n, k]^(k+2), {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Jan 16 2014 *)
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PROG
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(PARI) {a(n)=sum(k=0, n, binomial(n, k)^(k+2))}
for(n=0, 16, print1(a(n), ", "))
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CROSSREFS
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Cf. A206153 (exp), A184731, A206156, A206158, A206152, A167008, A220359.
Sequence in context: A066205 A113147 A335946 * A181445 A231969 A062499
Adjacent sequences: A206151 A206152 A206153 * A206155 A206156 A206157
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Feb 04 2012
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STATUS
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approved
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