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A322398 Triangle of the coefficients of Touchard's chord enumerating polynomials, [x^k] S(n,x), 0 <= k <= n*(n-1)/2.

%I A322398 #14 Jul 01 2023 16:03:28
%S A322398 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 A322398 42,210,540,945,1255,1353,1236,984,696,441,250,126,56,21,6,1,132,792,
%U A322398 2475,5335,8866,12112,14182,14654,13646,11619,9131,6662,4529,2870,1691,922,462,210,84,28,7,1,429,3003
%N A322398 Triangle of the coefficients of Touchard's chord enumerating polynomials, [x^k] S(n,x), 0 <= k <= n*(n-1)/2.
%H A322398 Jean-François Alcover, <a href="/A322398/b322398.txt">Table of n, a(n) for n = 1..1350 (20 rows)</a>
%H A322398 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 A322398 The triangle starts:
%e A322398     1;
%e A322398     1,    1;
%e A322398     2,    4,    3,    1;
%e A322398     5,   15,   21,   18,   10,    4,    1;
%e A322398    14,   56,  112,  148,  143,  109,   68,   35,   15,    5,    1;
%e A322398    ...
%p A322398 # page 3 prior to equation 2
%p A322398 Dpq := proc(p,q)
%p A322398     (p-q+1)*binomial(p+q,q)/(p+1) ;
%p A322398 end proc:
%p A322398 # page 12 top
%p A322398 fp1 := proc(p,x)
%p A322398     add( (-1)^i*Dpq(2*p-i,i)*x^((p+1-i)*(p-i)/2),i=0..p) ;
%p A322398 end proc:
%p A322398 # page 12
%p A322398 gnx := proc(n,x)
%p A322398     fp1(n,x)/(x-1)^n ;
%p A322398     taylor(%,x=0,1+n*(n+1)/2) ;
%p A322398     convert(%,polynom) ;
%p A322398 end proc:
%p A322398 Snx := proc(n,x)
%p A322398     if n =0 then
%p A322398         0;
%p A322398     elif n =1 then
%p A322398         1;
%p A322398     else
%p A322398         # recurrence page 17
%p A322398         gnx(n,x)-add( gnx(n-i,x)*procname(i,x),i=1..n-1) ;
%p A322398         taylor(%,x=1,1+n*(n+1)/2) ;
%p A322398         convert(%,polynom) ;
%p A322398         expand(%) ;
%p A322398     end if;
%p A322398 end proc:
%p A322398 for n from 1 to 8 do
%p A322398     S := Snx(n,x) ;
%p A322398     seq( coeff(S,x,i),i=0..n*(n-1)/2) ;
%p A322398     printf("\n") ;
%p A322398 end do:
%t A322398 Dpq[p_, q_] := (p - q + 1)*Binomial[p + q, q]/(p + 1);
%t A322398 fp1[p_, x_] := Sum[(-1)^i*Dpq[2*p - i, i]*x^((p + 1 - i)*(p - i)/2), {i, 0, p}];
%t A322398 gnx[n_, x_] := fp1[n, x]/(x - 1)^n // Series[#, {x, 0, 1 + n*(n + 1)/2}]& // Normal;
%t A322398 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 A322398 Table[CoefficientList[Snx[n, x], x], {n, 1, 8}] // Flatten (* _Jean-François Alcover_, Jul 01 2023, after _R. J. Mathar_ *)
%Y A322398 Cf. A000108 (leading column), A001791 (2nd column), A000698 (row sums).
%K A322398 nonn,tabf
%O A322398 1,4
%A A322398 _R. J. Mathar_, Dec 06 2018