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Search: id:A089946
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| A089946 |
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Secondary diagonal of array A089944, in which the n-th row is the n-th binomial transform of the natural numbers. |
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3
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| 1, 4, 24, 200, 2160, 28812, 458752, 8503056, 180000000, 4287177620, 113515167744, 3308603804376, 105288694411264, 3632897460937500, 135107988821114880, 5388090449900829728, 229385780960233586688, 10383890888434362036516
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Also the hyperbinomial transform of A089945 (the main diagonal of A089944): a(n)=sum(k=0,n,C(n,k)*(n-k+1)^(n-k-1)*A089945(k)).
With offset 1, a(n) = total number of children of the root in all (n+1)^(n-1) trees on {0,1,2,...,n} rooted at 0. For example, with edges directed away from the root, the trees on {0,1,2} are {0->1,0->2},{0->1->2},{0->2->1} and contain a total of a(2)=4 children of 0. - David Callan (callan(AT)stat.wisc.edu), Feb 01 2007
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FORMULA
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a(n)=2*(n+1)*(n+2)^(n-1). a(n)=sum(k=0, n, C(n, k)*(n-k+1)^(n-k-1)*(2*k+1)*(k+1)^(k-1)). E.g.f.: (-LambertW(x)/x)^2*(1-LambertW(x))/(1+LambertW(x)).
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PROGRAM
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(PARI) a(n)=if(n<0, 0, 2*(n+1)*(n+2)^(n-1));
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CROSSREFS
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Cf. A089944, A089945.
Adjacent sequences: A089943 A089944 A089945 this_sequence A089947 A089948 A089949
Sequence in context: A099021 A136229 A138419 this_sequence A012244 A050388 A010039
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Nov 23 2003
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