

A192020


Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the binomial tree of order n (1 <= k <= 2n1; entries in row n are the coefficients of the corresponding Wiener polynomial).


1



1, 3, 2, 1, 7, 8, 8, 4, 1, 15, 22, 31, 28, 17, 6, 1, 31, 52, 90, 112, 104, 68, 30, 8, 1, 63, 114, 225, 344, 418, 388, 270, 136, 47, 10, 1, 127, 240, 516, 908, 1331, 1568, 1464, 1064, 589, 240, 68, 12, 1, 255, 494, 1123, 2180, 3663, 5138, 5931, 5560, 4181, 2482, 1137, 388, 93, 14, 1
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OFFSET

0,2


COMMENTS

The binomial trees b(k) of order k are ordered trees defined as follows:
1. b(0) consists of a single node.
2. For k >= 1, b(k) is obtained from two copies of b(k1) by linking them in such a way that the root of one is the leftmost child of the root of the other. See the Iyer & Reddy references.
Row n contains 2n1 entries.
Kevin Ryde, Sep 14 2019: (Start)
In the formulas below, the generating function for number of vertices at depth is r(n,t) = (t+1)^n = Sum_{i=0..n} binomial(n,i)*t^i. The w(n,t) recurrence applied repeatedly is a sum of those, and from which the rational function for w(n,t).
T(n,k) as sum over j follows from which binomials are put at which indices in the g.f. Or the direct interpretation is to number vertices v=0 to 2^n1 inclusive with parent(v) = A129760(v) in the usual way, then suppose a pair of vertices u,v have their highest differing bit at position j, where j=1 as the least significant bit. One of u or v has a 1bit at j. To be distance k apart requires k1 further 1bits among the bits below j in u and v, hence binomial(2(j1),k1). The bits above j are the same in u and v and can be any 2^(nj) (those bits and 0's below are the common ancestor of u,v).
(End)


REFERENCES

K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, Wiener index of Binomial trees and Fibonacci trees, Int'l. J. Math. Engin. with Comp., Accepted for publication, Sept. 2009.
T. H. Cormen, C. E. Leiserson and R. L. Rivest: Introduction to Algorithms. MIT Press / McGrawHill (1990).


LINKS

Table of n, a(n) for n=0..63.
B. E. Sagan, YN. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959969.
K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, Wiener index of binomial trees and Fibonacci trees, arXiv:0910.4432 [cs.DM], 2009.


FORMULA

T(n,1) = A000225(n) = 2^n  1.
T(n,2) = A005803(n+1) = 2^(n+1)  2*n  2.
Sum_{k>=1} k*T(n,k) = A192021(n) (the Wiener indices).
The Wiener polynomial w(n,t) of the binomial tree of order n satisfies the recurrence relation w(n,t) = 2*w(n1,t) + t*(r(n1,t))^2, w(0,t)=0, where r(n,t) is the generating polynomial of the nodes of the binomial tree b(n) with respect to the level of the nodes (for example, r(1,t) = 1 + t for the oneedge tree b(1)=  ; see the Maple program).
T(n,k) = Sum_{j=1..n} 2^(nj)*binomial(2*j2, k1).
w(n,t) = Sum_{i=0..n1} 2^(n1i)*t*(t+1)^(2i) = t * ((t+1)^(2n)  2^n)/((t+1)^2  2).  Kevin Ryde, Sep 13 2019


EXAMPLE

T(2,1)=3, T(2,2)=2, T(2,3)=1 because the binomial tree b(2) is basically the path tree ABRC and we have 3 (AB, BR, RC), 2 (AR, BC), and 1 (AC) pairs of nodes at distances 1, 2, and 3, respectively.
Triangle starts:
1;
3, 2, 1;
7, 8, 8, 4, 1;
15, 22, 31, 28, 17, 6, 1;
31, 52, 90, 112, 104, 68, 30, 8, 1;


MAPLE

G := 1/(1zt*z): Gser := simplify(series(G, z = 0, 11)): for n from 0 to 8 do r[n] := sort(coeff(Gser, z, n)) end do: w[0] := 0: for n to 8 do w[n] := sort(expand(2*w[n1]+t*r[n1]^2)) end do: for n to 8 do seq(coeff(w[n], t, k), k = 1 .. 2*n1) end do; # yields sequence in triangular form


MATHEMATICA

max = 8; g = 1/(1  z  t*z); r = CoefficientList[ Series[g, {z, 0, max}], z]; w[0] = 0; w[n_] := w[n] = 2 w[n1] + t*r[[n]]^2; Flatten[ Table[ Drop[ CoefficientList[ w[n], t], 1], {n, 1, max}]] (* JeanFrançois Alcover, Oct 06 2011, after Maple *)


PROG

(PARI) a(n) = my(s=sqrtint(n), r=ns^2); sum(i=0, s, 2^(si)*binomial(2*i, r)); \\ Kevin Ryde, Sep 13 2019


CROSSREFS

Cf. A000225, A005803, A192021.
Sequence in context: A130462 A059380 A145035 * A171128 A122832 A056151
Adjacent sequences: A192017 A192018 A192019 * A192021 A192022 A192023


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Jun 22 2011


STATUS

approved



