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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.

%I A242431 #21 Feb 26 2021 18:37:05
%S A242431 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,
%T A242431 37,7,1,2048,1984,1498,730,230,50,8,1,8597,8469,7011,3939,1439,359,65,
%U A242431 9,1,38486,38230,33856,21568,9068,2588,530,82,10,1
%N 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.
%H A242431 Peter Luschny, <a href="/A242431/b242431.txt">Rows n = 0..50, flattened.</a>
%H A242431 Mathew Englander, <a href="/A089246/a089246.pdf">Comments on A101494 and A089246, and related sequences</a>
%F A242431 T(n, 0) = A047970(n).
%F A242431 Sum_{k=0..n} T(n, k) = A112532(n+1).
%F A242431 From _Mathew Englander_, Feb 25 2021: (Start)
%F A242431 T(n,k) = 1 + Sum_{i = k+1..n} i*(i+1)^(n-i).
%F A242431 T(n,k) = T(n,k+1) + (k+1)*(k+2)^(n-k-1) for 0 <= k < n.
%F A242431 T(n,k) = T(n,k+1) + (k+2)*(T(n-1,k) - T(n-1,k+1)) for 0 <= k <= n-2.
%F A242431 T(n,k) = Sum_{i = 0..n-k} (k+2)^i*A089246(n-k,i).
%F A242431 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)
%e A242431 0|    1;
%e A242431 1|    2,    1;
%e A242431 2|    5,    3,    1;
%e A242431 3|   14,   10,    4,   1;
%e A242431 4|   43,   35,   17,   5,   1;
%e A242431 5|  144,  128,   74,  26,   6,  1;
%e A242431 6|  523,  491,  329, 137,  37,  7, 1;
%e A242431 7| 2048, 1984, 1498, 730, 230, 50, 8, 1;
%p A242431 T := proc(n, k) option remember; local j;
%p A242431     if k=n then 1
%p A242431   elif k>n then 0
%p A242431   else (k+1)*T(n-1, k) + add(T(n-1, j), j=k..n)
%p A242431     fi end:
%p A242431 seq(print(seq(T(n,k), k=0..n)), n=0..7);
%o A242431 (Sage)
%o A242431 def A242431_rows():
%o A242431     T = []; n = 0
%o A242431     while True:
%o A242431         T.append(1)
%o A242431         yield T
%o A242431         for k in (0..n):
%o A242431             T[k] = (k+1)*T[k] + add(T[j] for j in (k..n))
%o A242431         n += 1
%o A242431 a = A242431_rows()
%o A242431 for n in range(8): next(a)
%Y A242431 Cf. A003101, A026898, A047969, A047970, A101494, A089246.
%K A242431 nonn,tabl
%O A242431 0,2
%A A242431 _Peter Luschny_, May 14 2014