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

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, 42, 210, 540, 945, 1255, 1353, 1236, 984, 696, 441, 250, 126, 56, 21, 6, 1, 132, 792, 2475, 5335, 8866, 12112, 14182, 14654, 13646, 11619, 9131, 6662, 4529, 2870, 1691, 922, 462, 210, 84, 28, 7, 1, 429, 3003

Offset: 1

Author

R. J. Mathar, Dec 06 2018

Keywords

• nonn
• tabf

Examples

			The triangle starts:
    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;
   ...
		

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..1350 (20 rows)
• J. Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25, S_n(x).

Crossrefs

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

Programs

Maple

# page 3 prior to equation 2
Dpq := proc(p,q)
    (p-q+1)*binomial(p+q,q)/(p+1) ;
end proc:
# page 12 top
fp1 := proc(p,x)
    add( (-1)^i*Dpq(2*p-i,i)*x^((p+1-i)*(p-i)/2),i=0..p) ;
end proc:
# page 12
gnx := proc(n,x)
    fp1(n,x)/(x-1)^n ;
    taylor(%,x=0,1+n*(n+1)/2) ;
    convert(%,polynom) ;
end proc:
Snx := proc(n,x)
    if n =0 then
        0;
    elif n =1 then
        1;
    else
        # recurrence page 17
        gnx(n,x)-add( gnx(n-i,x)*procname(i,x),i=1..n-1) ;
        taylor(%,x=1,1+n*(n+1)/2) ;
        convert(%,polynom) ;
        expand(%) ;
    end if;
end proc:
for n from 1 to 8 do
    S := Snx(n,x) ;
    seq( coeff(S,x,i),i=0..n*(n-1)/2) ;
    printf("\n") ;
end do:

Mathematica

Dpq[p_, q_] := (p - q + 1)*Binomial[p + q, q]/(p + 1);
fp1[p_, x_] := Sum[(-1)^i*Dpq[2*p - i, i]*x^((p + 1 - i)*(p - i)/2), {i, 0, p}];
gnx[n_, x_] := fp1[n, x]/(x - 1)^n // Series[#, {x, 0, 1 + n*(n + 1)/2}]& // Normal;
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];
Table[CoefficientList[Snx[n, x], x], {n, 1, 8}] // Flatten (* Jean-François Alcover, Jul 01 2023, after R. J. Mathar *)