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A143018 Triangle read by rows: T(n,k) (n >=2, k >=1) is the number of non-crossing connected graphs on n nodes on a circle such that the distance from a fixed node (root) to the next node is k. Rows are indexed 2,3,4,...; columns are indexed 1,2,3, ... . 3
1, 3, 1, 16, 6, 1, 105, 41, 9, 1, 768, 306, 75, 12, 1, 6006, 2422, 630, 118, 15, 1, 49152, 19980, 5394, 1104, 170, 18, 1, 415701, 169941, 47061, 10197, 1755, 231, 21, 1, 3604480, 1479786, 417439, 94116, 17425, 2610, 301, 24, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

Row sums yield A007297.

T(n,1) = A085614(n-1).

Sum_{k=1..n-1} k*T(n,k) = A143020(n).

LINKS

Andrew Howroyd, Table of n, a(n) for n = 2..1276

P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.

FORMULA

T(n,k) = k*L(n-k-1,3n-k-4,n-1)/(n-1) (n>=2, 1<=k<=n--1), where L(p,q,r)=[u^p](1+u)^q/(1-u)^r = Sum[binom(q,i)binom(r+p-1-i,r-1), i=0..min(p,q)].

G.f.: G(t,z)=zg/[g - t(g - z)], where g=g(z), the g.f. for the number of non-crossing connected graphs on n nodes on a circle, satisfies g^3 + g^2 -3zg +2z^2=0 (A007297).

T(n,k) = k*Sum_{i=0..min(n-k-1, 3*n-k-4)} binomial(3*n-k-4, i)*binomial(2*n-k-i-3, n-2)/(n-1). - Andrew Howroyd, Nov 17 2017

EXAMPLE

T(3,1)=3 and T(3,2)=1 because in the graphs (AB,BC,CA), (AB,AC), (AB,BC) and (AC,BC) the distances from A to B are 1, 1, 1 and 2, respectively.

Triangle starts:

    1;

    3,   1;

   16,   6,  1;

  105,  41,  9,  1;

  768, 306, 75, 12, 1;

  ...

MAPLE

L:=proc(p, q, r) options operator, arrow: sum(binomial(q, i)*binomial(r+p-1-i, r-1), i=0..min(p, q)) end proc: T:=proc(n, k) options operator, arrow: k*L(n-k-1, 3*n-k-4, n-1)/(n-1) end proc: for n from 2 to 10 do seq(T(n, k), k=1..n-1) end do; # yields sequence in triangular form

MATHEMATICA

t[n_, k_] := k*L[n - k - 1, 3*n - k - 4, n-1]/(n-1); L[p_, q_, r_] := Sum[ Binomial[q, i]*Binomial[r + p - 1 - i, r-1], {i, 0, Min[p, q]}]; Flatten[ Table[ t[n, k], {n, 2, 10}, {k, 1, n-1}]] (* Jean-Fran├žois Alcover, Oct 05 2011, Oct 05 2011, after Maple *)

PROG

(PARI)

T(n, k)=k*sum(i=0, min(n-k-1, 3*n-k-4), binomial(3*n-k-4, i)*binomial(2*n-k-i-3, n-2))/(n-1);

for(n=2, 10, for(k=1, n-1, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 17 2017

CROSSREFS

Cf. A085614, A143020, A007297.

Sequence in context: A160616 A168319 A143565 * A102012 A128249 A071211

Adjacent sequences:  A143015 A143016 A143017 * A143019 A143020 A143021

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Jul 30 2008

STATUS

approved

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Last modified January 18 06:34 EST 2019. Contains 319269 sequences. (Running on oeis4.)