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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. 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Table of n, a(n) for n=1..65.

J. Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25, S_n(x).

EXAMPLE

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;

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:

CROSSREFS

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

Sequence in context: A011170 A304337 A274329 * A341606 A109158 A307500

Adjacent sequences:  A322395 A322396 A322397 * A322399 A322400 A322401

KEYWORD

nonn,tabf

AUTHOR

R. J. Mathar, Dec 06 2018

STATUS

approved

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Last modified June 26 20:07 EDT 2022. Contains 354885 sequences. (Running on oeis4.)