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A108704
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Number of partitions of 112233...nn into n pairs.
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1
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1, 1, 4, 18, 126, 1110, 12120, 156660, 2341500, 39701340, 752839920, 15785181720, 362606123880, 9055825538760, 244296192460320, 7079382509799600, 219321853964413200, 7233629128601475600, 253054306933115688000, 9358989706213886138400, 364860828050107348159200
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OFFSET
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0,3
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REFERENCES
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Laszlo Lovasz, Combinatorial Problems and Solutions, AMS Chelsea Publishing, American Mathematical Society.
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LINKS
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FORMULA
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E.g.f.: exp(x*x/2)/sqrt(1-2*x).
a(n) = 2^n*(n-1/2)!*2F2((1-n)/2,-n/2;1/4 -n/2,3/4 - n/2; 1/8)/sqrt(Pi). - Benedict W. J. Irwin, May 25 2016
Conjecture: a(n)-(2*n-1)*a(n-1)-(n-1)*a(n-2)+2*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Jun 08 2016
The exponential generating series A(t) satisfies the differential equation (1-2*t)*A'(t) = (1+t-2*t^2)*A(t), which is equivalent to the conjectured recurrence.
a(n) = Sum_{k=0..n/2} binomial(n,k)*binomial(2*n-4*k,n-2*k)*(n-k)!/2^(n-k).
(End)
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EXAMPLE
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Partitions of 1122 into 2 pairs: 11 22, 12 12, 12 21, 21 21 = 4 partitions so a(2)=4.
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MAPLE
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a:= n-> n! *coeff(series(exp(x*x/2)/sqrt(1-2*x), x, n+1), x, n):
seq (a(n), n=0..20);
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MATHEMATICA
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CoefficientList[Series[E^(x*x/2)/Sqrt[1-2*x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 26 2013 *)
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PROG
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(PARI) my(x='x+O('x^50)); Vec(serlaplace(exp(x*x/2)/sqrt(1-2*x))) \\ G. C. Greubel, May 24 2017
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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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