OFFSET
1,4
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).
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:
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 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
R. J. Mathar, Dec 06 2018
STATUS
approved