A322398 - OEIS

%I #14 Jul 01 2023 16:03:28

%S 1,1,1,2,4,3,1,5,15,21,18,10,4,1,14,56,112,148,143,109,68,35,15,5,1,

%T 42,210,540,945,1255,1353,1236,984,696,441,250,126,56,21,6,1,132,792,

%U 2475,5335,8866,12112,14182,14654,13646,11619,9131,6662,4529,2870,1691,922,462,210,84,28,7,1,429,3003

%N Triangle of the coefficients of Touchard's chord enumerating polynomials, [x^k] S(n,x), 0 <= k <= n*(n-1)/2.

%H Jean-François Alcover, <a href="/A322398/b322398.txt">Table of n, a(n) for n = 1..1350 (20 rows)</a>

%H J. Touchard, <a href="http://dx.doi.org/10.4153/CJM-1952-001-8">Sur un problème de configurations et sur les fractions continues</a>, Canad. J. Math., 4 (1952), 2-25, S_n(x).

%e The triangle starts:

%e     1;

%e     1,    1;

%e     2,    4,    3,    1;

%e     5,   15,   21,   18,   10,    4,    1;

%e    14,   56,  112,  148,  143,  109,   68,   35,   15,    5,    1;

%e    ...

%p # page 3 prior to equation 2

%p Dpq := proc(p,q)

%p     (p-q+1)*binomial(p+q,q)/(p+1) ;

%p end proc:

%p # page 12 top

%p fp1 := proc(p,x)

%p     add( (-1)^i*Dpq(2*p-i,i)*x^((p+1-i)*(p-i)/2),i=0..p) ;

%p end proc:

%p # page 12

%p gnx := proc(n,x)

%p     fp1(n,x)/(x-1)^n ;

%p     taylor(%,x=0,1+n*(n+1)/2) ;

%p     convert(%,polynom) ;

%p end proc:

%p Snx := proc(n,x)

%p     if n =0 then

%p         0;

%p     elif n =1 then

%p         1;

%p     else

%p         # recurrence page 17

%p         gnx(n,x)-add( gnx(n-i,x)*procname(i,x),i=1..n-1) ;

%p         taylor(%,x=1,1+n*(n+1)/2) ;

%p         convert(%,polynom) ;

%p         expand(%) ;

%p     end if;

%p end proc:

%p for n from 1 to 8 do

%p     S := Snx(n,x) ;

%p     seq( coeff(S,x,i),i=0..n*(n-1)/2) ;

%p     printf("\n") ;

%p end do:

%t Dpq[p_, q_] := (p - q + 1)*Binomial[p + q, q]/(p + 1);

%t fp1[p_, x_] := Sum[(-1)^i*Dpq[2*p - i, i]*x^((p + 1 - i)*(p - i)/2), {i, 0, p}];

%t gnx[n_, x_] := fp1[n, x]/(x - 1)^n // Series[#, {x, 0, 1 + n*(n + 1)/2}]& // Normal;

%t Snx[n_, x_] := Snx[n, x] = Which[n == 0, 0, n == 1, 1, True, gnx[n, x] - Sum[gnx[n - i, x]*Snx[i, x], {i, 1, n - 1}] // Series[#, {x, 1, 1 + n*(n + 1)/2}]& // Normal];

%t Table[CoefficientList[Snx[n, x], x], {n, 1, 8}] // Flatten (* _Jean-François Alcover_, Jul 01 2023, after _R. J. Mathar_ *)

%Y Cf. A000108 (leading column), A001791 (2nd column), A000698 (row sums).

%K nonn,tabf

%O 1,4

%A _R. J. Mathar_, Dec 06 2018