OFFSET
0,8
COMMENTS
Conjecture: Let A = (g(t), f(t)) and B = (u(t), v(t)) be (triangular) Riordan arrays with A(n, k) = [t^n](g(t)*(f(t))^k) and B(n, k) = [t^n](u(t)*(v(t))^k). Then T = (g(t)*u(f(t)), v(f(t))*t/f(t)) is the Riordan array with T(n, k) = [t^n](g(t)*u(f(t))*(v(f(t))*t/f(t))^k) = Sum_{i=0..n-k} A(n-k, i) * B(k+i, k) for 0 <= k <= n.
FORMULA
Riordan array (C(t/(1+t)) / (1+t), t * C(t/(1+t))) where C(x) is g.f. of A000108.
Riordan array ((1 + t - sqrt(1 - 2*t - 3*t^2))/(2*t*(1 + t)), (1 + t - sqrt(1-2*t-3*t^2))/2).
G.f.: 2/(sqrt((1 - 3*t)*(t + 1)) - 2*(t + 1)*t*x + t + 1).
Conjecture: T(n, k) = T(n, k-1) + T(n-1, k-1) - T(n-1, k-2) - T(n-2, k-2) for 2 <= k <= n.
T(n, k) = (-1)^(k-n)*hypergeom([k-n, k/2+1, (k+1)/2], [1, k + 2], 4). - Peter Luschny, Jan 06 2025
EXAMPLE
Triangle T(n, k) for 0 <= k <= n starts:
n \k : 0 1 2 3 4 5 6 7 8 9 10 11
====================================================================
0 : 1
1 : 0 1
2 : 1 1 1
3 : 1 2 2 1
4 : 3 4 4 3 1
5 : 6 9 9 7 4 1
6 : 15 21 21 17 11 5 1
7 : 36 51 51 42 29 16 6 1
8 : 91 127 127 106 76 46 22 7 1
9 : 232 323 323 272 200 128 69 29 8 1
10 : 603 835 835 708 530 352 204 99 37 9 1
11 : 1585 2188 2188 1865 1415 965 587 311 137 46 10 1
etc.
MAPLE
gf := 2/(sqrt((1-3*t)*(t+1)) - 2*(t+1)*t*x + t+1): ser := simplify(series(gf, t, 12)):
ct := n -> coeff(ser, t, n): row := n -> local k; seq(coeff(ct(n), x, k), k = 0..n):
seq(row(n), n = 0..11); # Peter Luschny, Jan 05 2025
PROG
(PARI) T(n, k) = sum(i=0, n-k, (-1)^(n-k-i)*binomial(n-k, i)*binomial(k+2*i, i)*(k+1)/(k+1+i))
(PARI) T(n, k)=polcoef(polcoef(2/(sqrt((1-3*t)*(1+t))+(1+t)*(1-2*x*t))+x*O(x^k), k, x)+t*O(t^n), n, t);
m=matrix(15, 15, n, k, if(k>n, 0, T(n-1, k-1)))
CROSSREFS
KEYWORD
AUTHOR
Werner Schulte, Jan 05 2025
STATUS
approved