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A072678
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Generalized Bell numbers B_{4,2}.
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2
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1, 21, 1045, 93289, 12975561, 2581284541, 693347907421, 241253367679185, 105394372192969489, 56410454014314490981, 36271084122927079387941, 27567930377271475039277881, 24435533594428382909107147225
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = (2*n)!*hypergeom([2*n+1], [3], 1)/(2*exp(1)), n=1, 2, ... Special values of the confluent hypergeometric function 1F1.
a(n) = sum(A090438(n, k), k=2..2*n)= sum((1/k!)*product(fallfac(k+(j-1)*(4-2), 2), j=1..n), k=1..infinity)/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=4, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.
8*n*(2*n-1)*(2*n+1)*(n+1)^2*(n+3)*(n+2)*a(n)+(2*(n+1))*(8*n^3+32*n^2+42*n+13)*a(n+1)*(n+3)*(n+2)-(8*n^2+38*n+51)*(n+3)*(n+2)*a(n+2)+(n+3)*(n+2)*a(n+3) = 0. - Robert Israel, May 23 2016
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MAPLE
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f:= n -> simplify((2*n)!*hypergeom([2*n+1], [3], 1)/(2*exp(1))):
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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