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

0,5

Links

Table of n, a(n) for n=0..65.

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.

Example

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.

Prog

(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)))

Crossrefs

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

Sequence in context: A339549 A179750 A091173 * A101897 A208058 A078142

Adjacent sequences: A378142 A378143 A378144 * A378146 A378147 A378148

Keyword

nonn,easy,tabl

Author

Werner Schulte, Nov 17 2024

Status

approved