A192024 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the double-comb graph \/_\/_\/...\/_\/ with 3n (n>=1) nodes. The entries in row n are the coefficients of the corresponding Wiener polynomial.

2, 1, 5, 6, 4, 8, 12, 12, 4, 11, 18, 21, 12, 4, 14, 24, 30, 21, 12, 4, 17, 30, 39, 30, 21, 12, 4, 20, 36, 48, 39, 30, 21, 12, 4, 23, 42, 57, 48, 39, 30, 21, 12, 4, 26, 48, 66, 57, 48, 39, 30, 21, 12, 4, 29, 54, 75, 66, 57, 48, 39, 30, 21, 12, 4, 32, 60, 84, 75, 66, 57, 48, 39, 30, 21, 12, 4

Offset

1,1

Comments

Row n contains n+1 entries.

Sum of entries in row n is 3*n*(3*n-1)/2=A062741(n).

Sum(k*T(n,k),k>=1)=A192025(n) (the Wiener indices).

Links

Table of n, a(n) for n=1..77.

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Formula

G.f.: G(t,z)=t*z*(2+t+z+2*t*z+3*t^2*z)/((1-t*z)*(1-z)^2).

G.f. of column 1: z*(2+z)/(1-z)^2.

G.f. of column 2: z*(1+4*z+z^2)/(1-z)^2.

G.f. of column k>=3: z^(k-1)*(4+4*z+z^2)/(1-z)^2.

The generating polynomial of row n (i.e. the Wiener polynomial of the double-comb with 3n nodes) is n*(2*t +t^2)+t*(1+2*t)^2*(n*(1-t)-(1-t^n))/(1-t)^2 or, equivalently, n*(2*t+t^2)+t*(1+2*t)^2*Sum((n-j)*t^(j-1), j=1..n-1).

Example

T(2,1)=5, T(2,2)=6, T(2,3)=4 because in the graph \/_\/ there are 5 pairs of nodes at distance 1, 6 pairs at distance 2, and 4 pairs at distance 3.
Triangle starts:
2,1;
5,6,4;
8,12,12,4;
11,18,21,12,4;

Maple

Q := proc (n) options operator, arrow: n*(t^2+2*t)+t*(1+2*t)^2*(sum((n-j)*t^(j-1), j = 1 .. n-1)) end proc: for n to 11 do P[n] := sort(expand(Q(n))) end do: for n to 11 do seq(coeff(P[n], t, j), j = 1 .. n+1) end do; # yields sequence in triangular form

Crossrefs

Cf. A062741, A192025

Sequence in context: A330984 A353648 A075680 * A249283 A176035 A190992

Adjacent sequences: A192021 A192022 A192023 * A192025 A192026 A192027

Keyword

nonn,tabf

Author

Emeric Deutsch, Jun 25 2011

Status

approved