A378145 Riordan triangle (1 + x * C(x), x * C(x)), where C(x) is g.f. of A000108.

1, 1, 1, 1, 2, 1, 2, 4, 3, 1, 5, 10, 8, 4, 1, 14, 28, 23, 13, 5, 1, 42, 84, 70, 42, 19, 6, 1, 132, 264, 222, 138, 68, 26, 7, 1, 429, 858, 726, 462, 240, 102, 34, 8, 1, 1430, 2860, 2431, 1573, 847, 385, 145, 43, 9, 1, 4862, 9724, 8294, 5434, 3003, 1430, 583, 198, 53, 10, 1

Offset: 0

Author

Werner Schulte, Nov 17 2024

Keywords

• nonn
• easy
• tabl

Examples

			Triangle T(n, k) for 0 <= k <= n starts:
n\k :     0     1     2     3    4    5    6   7  8  9
======================================================
  0 :     1
  1 :     1     1
  2 :     1     2     1
  3 :     2     4     3     1
  4 :     5    10     8     4    1
  5 :    14    28    23    13    5    1
  6 :    42    84    70    42   19    6    1
  7 :   132   264   222   138   68   26    7   1
  8 :   429   858   726   462  240  102   34   8  1
  9 :  1430  2860  2431  1573  847  385  145  43  9  1
  etc.
		

Crossrefs

Cf. A000108, A004070, A120588 (column 0), A068875 (column 1 and row sums), A000007 (alt. row sums).

Programs

PARI

T(n,k)=if(k==n,1,binomial(2*n-k,n)*(n*(3*k+1)-2*k*(k+1))/((2*n-k)*(2*n-k-1)))

Formula

T(n, k) = binomial(2*n-k, n) * (n*(3*k+1) - 2*k*(k+1)) / ((2*n-k) * (2*n-k-1)) if 0 <= k < n and 1 if k = n.

T(n, k) = T(n, k-1) - T(n-1, k-2) for 2 <= k <= n.

(-1)^(n-k) * T(n, k) is matrix inverse of A004070 (seen as a triangle).

Conjecture: Sum_{i=0..n-k} binomial(i+m-1, i) * T(n, i+k) = T(n+m, m+k) for m > 0.

Conjecture: Sum_{k=0..n} (1 + floor(k/2)) * T(n, k) = A000108(n+1).

G.f.: A(x, y) = (1 + x*C(x)) / (1 - y * x*C(x)), where C(x) is g.f. of A000108.