A379907 Triangle read by rows: 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).

1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 3, 1, 6, 9, 9, 7, 4, 1, 15, 21, 21, 17, 11, 5, 1, 36, 51, 51, 42, 29, 16, 6, 1, 91, 127, 127, 106, 76, 46, 22, 7, 1, 232, 323, 323, 272, 200, 128, 69, 29, 8, 1, 603, 835, 835, 708, 530, 352, 204, 99, 37, 9, 1, 1585, 2188, 2188, 1865, 1415, 965, 587, 311, 137, 46, 10, 1

Offset: 0

Author

Werner Schulte, Jan 05 2025

Keywords

• nonn
• easy
• tabl

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.

Examples

			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.
		

Crossrefs

Cf. A005043 (column 0), A001006 (column 1 and 2), A102071 (column 3).

Cf. A000108, A342912 (row sums), A379824 (alternating row sums), A379823 (central terms).

Programs

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

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)))

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