A242431 Triangle read by rows: T(n, k) = (k + 1)*T(n-1, k) + Sum_{j=k..n-1} T(n-1, j) for k < n, T(n, n) = 1. T(n, k) for n >= 0 and 0 <= k <= n.

1, 2, 1, 5, 3, 1, 14, 10, 4, 1, 43, 35, 17, 5, 1, 144, 128, 74, 26, 6, 1, 523, 491, 329, 137, 37, 7, 1, 2048, 1984, 1498, 730, 230, 50, 8, 1, 8597, 8469, 7011, 3939, 1439, 359, 65, 9, 1, 38486, 38230, 33856, 21568, 9068, 2588, 530, 82, 10, 1

Offset: 0

Author

Peter Luschny, May 14 2014

Keywords

• nonn
• tabl

Examples

			0|    1;
1|    2,    1;
2|    5,    3,    1;
3|   14,   10,    4,   1;
4|   43,   35,   17,   5,   1;
5|  144,  128,   74,  26,   6,  1;
6|  523,  491,  329, 137,  37,  7, 1;
7| 2048, 1984, 1498, 730, 230, 50, 8, 1;
		

Links

• Peter Luschny, Rows n = 0..50, flattened.
• Mathew Englander, Comments on A101494 and A089246, and related sequences

Crossrefs

Cf. A003101, A026898, A047969, A047970, A101494, A089246.

Programs

Maple

T := proc(n, k) option remember; local j;
    if k=n then 1
  elif k>n then 0
  else (k+1)*T(n-1, k) + add(T(n-1, j), j=k..n)
    fi end:
seq(print(seq(T(n,k), k=0..n)), n=0..7);

Sage

def A242431_rows():
    T = []; n = 0
    while True:
        T.append(1)
        yield T
        for k in (0..n):
            T[k] = (k+1)*T[k] + add(T[j] for j in (k..n))
        n += 1
a = A242431_rows()
for n in range(8): next(a)

Formula

T(n, 0) = A047970(n).

Sum_{k=0..n} T(n, k) = A112532(n+1).

From Mathew Englander, Feb 25 2021: (Start)

T(n,k) = 1 + Sum_{i = k+1..n} i*(i+1)^(n-i).

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

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

T(n,k) = Sum_{i = 0..n-k} (k+2)^i*A089246(n-k,i).

Sum_{i = k..n} T(i,k) = Sum_{i = 0..n-k} (n+2-i)^i = Sum_{i = 0..n-k} A101494(n-k,i)*(k+2)^i. (End)