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A301741
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a(n) = n! * [x^n] exp((n + 1)*x + x^2/2).
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1
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1, 2, 10, 76, 778, 10026, 155884, 2839880, 59339004, 1399069450, 36746349496, 1064024248068, 33676500286840, 1156685567791586, 42850609041047760, 1703182952266379536, 72299420602524921616, 3264579136056004359570, 156238968782480840396704, 7900247992586138688381500
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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) = [x^n] 1/(1 - (n + 1)*x - x^2/(1 - (n + 1)*x - 2*x^2/(1 - (n + 1)*x - 3*x^2/(1 - ...)))), a continued fraction.
a(n) = Sum_{k=0..floor(n/2)} n!*(n + 1)^(n-2*k)/(2^k*k!*(n - 2*k)!).
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MATHEMATICA
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Table[n! SeriesCoefficient[Exp[(n + 1) x + x^2/2], {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[1/(1 - (n + 1) x + ContinuedFractionK[-k x^2, 1 - (n + 1) x, {k, 1, n}]), {x, 0, n}], {n, 0, 19}]
Table[Sum[n! (n + 1)^(n - 2 k)/(2^k k! (n - 2 k)!), {k, 0, Floor[n/2]}], {n, 0, 19}]
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PROG
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(GAP) List([0..10], n->Sum([0..Int(n/2)], k->Factorial(n)*(n+1)^(n-2*k)/(2^k*Factorial(k)*Factorial(n-2*k)))); # Muniru A Asiru, Mar 26 2018
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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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