A192022 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the comb-shaped graph |_|_|...|_| with 2n (n>=1) nodes. The entries in row n are the coefficients of the corresponding Wiener polynomial.

1, 0, 3, 2, 1, 5, 5, 4, 1, 7, 8, 8, 4, 1, 9, 11, 12, 8, 4, 1, 11, 14, 16, 12, 8, 4, 1, 13, 17, 20, 16, 12, 8, 4, 1, 15, 20, 24, 20, 16, 12, 8, 4, 1, 17, 23, 28, 24, 20, 16, 12, 8, 4, 1, 19, 26, 32, 28, 24, 20, 16, 12, 8, 4, 1, 21, 29, 36, 32, 28, 24, 20, 16, 12, 8, 4, 1, 23, 32, 40, 36, 32, 28, 24, 20, 16, 12, 8, 4, 1

Offset

1,3

Comments

Row n>=2 contains n+1 entries.

Sum of entries in row n is n*(2n-1)=A000384(n) (the hexagonal numbers).

Sum(k*T(n,k),k>=1)=A192023(n) (the Wiener indices).

Links

Table of n, a(n) for n=1..90.

Formula

The generating polynomial of row n (i.e. the Wiener polynomial of the comb with 2n nodes) is n*t + t*(1+t)^2*(n*(1-t)-(1-t^n))/(1-t)^2 or, equivalently, n*t + t*(1+t)^2*Sum((n-j)*t^(j-1),j=1..n-1).

Example

T(2,1)=3, T(2,2)=2, T(2,3)=1 because in the graph |_| there are 3 pairs of nodes at distance 1, 2 pairs at distance 2, and 1 pair at distance 3.
Triangle starts:
1,0;
3,2,1;
5,5,4,1;
7,8,8,4,1;
9,11,12,8,4,1;

Maple

Q := proc (n) options operator, arrow: n*t+t*(1+t)^2*(sum((n-j)*t^(j-1), j = 1 .. n-1)) end proc: for n to 12 do P[n] := sort(expand(Q(n))) end do: 1; for n from 2 to 12 do seq(coeff(P[n], t, j), j = 1 .. n+1) end do; # yields sequence in triangular form

Crossrefs

Cf. A000384, A192023

Sequence in context: A246837 A132969 A132970 * A208608 A209577 A139377

Adjacent sequences: A192019 A192020 A192021 * A192023 A192024 A192025

Keyword

nonn,tabf

Author

Emeric Deutsch, Jun 24 2011

Status

approved