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A143981
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The number of unigraphical partitions of 2m; that is, the number of partitions of 2m which are realizable as the degree sequence of one and only one graph (where loops are not allowed but multiple edges are allowed).
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0
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1, 3, 6, 9, 15, 19, 26, 36, 46, 59, 80, 100, 128, 167, 211, 267, 341, 429, 541, 682, 850, 1063, 1327, 1647, 2035, 2520, 3100, 3810, 4669, 5708, 6955, 8468, 10267, 12441, 15026, 18120, 21788, 26175, 31355, 37510, 44769, 53362, 63460, 75384, 89348
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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REFERENCES
| S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph. I, J. Soc. Indust. Appl. Math., vol. 10 (1962), 496-506
S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph. II. Uniqueness, J. Soc. Indust. Appl. Math., vol. 11 (1963), 135-147
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FORMULA
| For m >= 3, a(2m) = A000041(m) + A001399(m-3) + A000005(m+1) + A083039(m-2) + m - 5
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EXAMPLE
| For m = 4, the number of unigraphical partitions is A000041(4) + A001399(1) + A000005(5) + A083039(2) + 4 - 5 = 5 + 1 + 2 + 2 + 4 - 5 = 9.
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MAPLE
| with(combinat): with(numtheory): a:=proc(m) it:=round(m^2/12)+numbpart(m)+tau(m+1)+m-5: if m mod 6 = 0 then it:=it+2 fi: if m mod 6 = 1 then it:=it+1 fi: if m mod 6 = 2 then it:=it+3 fi: if m mod 6 = 3 then it:=it+1 fi: if m mod 6 = 4 then it:=it+2 fi: if m mod 6 = 5 then it:=it+2 fi: RETURN(it): end:
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CROSSREFS
| Sequence in context: A000741 A133205 A049991 * A031940 A007187 A082004
Adjacent sequences: A143978 A143979 A143980 * A143982 A143983 A143984
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KEYWORD
| nonn
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AUTHOR
| Michael D. Hirschhorn and James A. Sellers (sellersj(AT)math.psu.edu), Sep 06 2008
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