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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 J. Touchard, Sur un problĂ¨me 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.)