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

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Last modified March 27 12:57 EDT 2023. Contains 361570 sequences. (Running on oeis4.)