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A192031
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Rectangular array read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the helm graph G(n) obtained from a wheel graph with n spokes by adjoining a pendant edge at each node of the cycle (n>=3, k>=1). The entries in row n are the coefficients of the corresponding Wiener polynomial.
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0
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9, 9, 3, 12, 14, 8, 2, 15, 20, 15, 5, 18, 27, 24, 9, 21, 35, 35, 14, 24, 44, 48, 20, 27, 54, 63, 27, 30, 65, 80, 35, 33, 77, 99, 44, 36, 90, 120, 54, 39, 104, 143, 65, 42, 119, 168, 77, 45, 135, 195, 90, 48, 152, 224, 104, 51, 170, 255, 119, 54, 189, 288, 135, 57, 209, 323, 152, 60, 230, 360, 170
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OFFSET
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3,1
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COMMENTS
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The graph G(n) is a special case of the graph G(n,m) defined in A192026 (m=1).
T(n,k) is also the number of unordered pairs of nodes at distance k in the gear graph G(n) obtained from a wheel graph with n spokes by adding a node between each pair of adjacent nodes of the cycle (n>=3, k>=1). Example: T(3,3)=3 because in the graph G(3) with vertex set {A,B,C,D,B',C',D'} and edge set {BD', D'C, CB', B'D, DC', C'B,AB,AC,AD} there are exactly 3 pairs of vertices at distance 3: BB', CC', and DD'.
Row 3 contains 3 entries; row n>=4 contains 4 entries.
Sum of entries in row n is n*(2n+1)=A014105(n).
Sum(k*T(n,k),k>=1)=6*n*(n-1)=A049598(n-1) (the Wiener indices).
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LINKS
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Eric Weisstein's World of Mathematics, Helm Graph.
Eric Weisstein's World of Mathematics, Gear Graph.
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FORMULA
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Generating polynomial of row n (i.e. the Wiener polynomial of the graph G(n)) is P(n;t)=3*n*t+(1/2)*n*(n+3)*t^2+n*(n-2)*t^3+(1/2)*n*(n-3)*t^4.
G.f.: x^3*(9 + 3*x^2 + 9*x^3 - 22*x^4 + 3*x^5 - 10*x^6 - 11*x^7 + 16*x^8 + 2*x^9 + 6*x^10 + 3*x^11 - 3*x^12 - 3*x^13) / ((1 - x)^3*(1 + x)^2*(1 + x^2)^3).
a(n) = a(n-1) - a(n-2) + a(n-3) + 2*a(n-4) - 2*a(n-5) + 2*a(n-6) - 2*a(n-7) - a(n-8) + a(n-9) - a(n-10) + a(n-11) for n>16.
(End)
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EXAMPLE
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T(3,3)=3 because in the graph G(3) with vertex set {A,B,C,D,B',C',D'} and edge set {BC,CD,DB,AB,AC,AD,BB',CC",DD'} there are exactly 3 pairs of vertices at distance 3: B'C', C'D', and D'B'.
Rectangular array starts:
9,9,3;
12,14,8,2;
15,20,15,5;
18,27,24,9;
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MAPLE
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P := proc (n) options operator, arrow: 3*n*t+(1/2)*n*(n+3)*t^2+n*(n-2)*t^3+(1/2)*n*(n-3)*t^4 end proc: T := proc (n, k) options operator, arrow: coeff(P(n), t, k) end proc: seq(T(3, k), k = 1 .. 3); for n from 4 to 20 do seq(T(n, k), k = 1 .. 4) end do; # yields rows 3, 4, ..., 20 of the rectangular array
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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