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A242431
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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.
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3
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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LINKS
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Peter Luschny, Rows n = 0..50, flattened.
Mathew Englander, Comments on A101494 and A089246, and related sequences
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FORMULA
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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)
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EXAMPLE
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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;
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MAPLE
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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);
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PROG
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(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)
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CROSSREFS
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Cf. A003101, A026898, A047969, A047970, A101494, A089246.
Sequence in context: A105848 A048471 A067345 * A349934 A188416 A160185
Adjacent sequences: A242428 A242429 A242430 * A242432 A242433 A242434
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KEYWORD
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nonn,tabl
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AUTHOR
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Peter Luschny, May 14 2014
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STATUS
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approved
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