A344381 Triangle read by rows: T(n,k) = (n+2) * (Sum_{i=k..n} i!) / ((k+2) * k!) for 0 <= k <= n with T(i,j) = 0 if j < 0 or i < j.

1, 3, 1, 8, 4, 1, 25, 15, 5, 1, 102, 66, 24, 6, 1, 539, 357, 133, 35, 7, 1, 3496, 2328, 872, 232, 48, 8, 1, 26613, 17739, 6651, 1773, 369, 63, 9, 1, 231170, 154110, 57790, 15410, 3210, 550, 80, 10, 1, 2250127, 1500081, 562529, 150007, 31251, 5357, 781, 99, 11, 1

Offset

0,2

Comments

There is a relationship to the left factorials A003422 (see formula).

Links

Table of n, a(n) for n=0..54.

Formula

T(n,k) = (n+2) * (A003422(n+1)-A003422(k)) / ((k+2) * k!) for 0 <= k <= n.

T(n,n) = 1 for n >= 0; T(n,n-1) = n+2 for n > 0.

T(n,k) = (n+2) * (T(n-1,k) - T(n-2,k)) for 0 < k+1 < n.

T(n,k) + T(n-2,k) - 2 * T(n-1,k) = (n+1)! / ((k+2) * k!) for 0 < k+1 < n.

T(n,k) * T(n-1,k-1) - T(n,k-1) * T(n-1,k) = (n+2)! / (k+2)! for 0 < k < n.

The row polynomials p(n; x) = Sum_{k=0..n} T(n,k) * x^k satisfy the recurrence equation p(n; x) = (n+2) * (p(n-1; x) - p(n-2; x)) + x^n for n > 1 with p(0; x) = 1 and p(1; x) = 3 + x.

Row sums are p(n; 1): 1, 4, 13, 46, 199, 1072, ....

Alternating row sums are p(n; -1): 1, 2, 5, 14, 55, 286, ....

With T as lower triangular matrix the matrix inverse M is defined by: M(n,n) = 1 for n >= 0, M(n,n-1) = -(n+2) for n > 0, M(n,n-2) = n+2 for n > 1, and M(i,j) = 0 otherwise.

Example

The triangle T(n,k), 0 <= k <= n, begins:
n\k :        0        1       2       3      4     5    6   7   8  9
====================================================================
  0 :        1
  1 :        3        1
  2 :        8        4       1
  3 :       25       15       5       1
  4 :      102       66      24       6      1
  5 :      539      357     133      35      7     1
  6 :     3496     2328     872     232     48     8    1
  7 :    26613    17739    6651    1773    369    63    9   1
  8 :   231170   154110   57790   15410   3210   550   80  10   1
  9 :  2250127  1500081  562529  150007  31251  5357  781  99  11  1
  etc.
The matrix inverse M(n,k), 0 <= k <= n, begins:
n\k :   0   1   2   3   4   5    6    7    8  9
===============================================
  0 :   1
  1 :  -3   1
  2 :   4  -4   1
  3 :   0   5  -5   1
  4 :   0   0   6  -6   1
  5 :   0   0   0   7  -7   1
  6 :   0   0   0   0   8  -8    1
  7 :   0   0   0   0   0   9   -9    1
  8 :   0   0   0   0   0   0   10  -10    1
  9 :   0   0   0   0   0   0    0   11  -11  1
  etc.

Mathematica

T[n_, k_] := (n + 2) * Sum[i!, {i, k, n}]/((k + 2)*k!); Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 12 2021 *)

Crossrefs

Cf. A003422, A051403 (column 0).

Sequence in context: A101026 A055249 A125172 * A073732 A021318 A068437

Adjacent sequences: A344378 A344379 A344380 * A344382 A344383 A344384

Keyword

nonn,easy,tabl

Author

Werner Schulte, May 16 2021

Status

approved