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A201634 Triangle read by rows, n>=0, k>=0, T(n,n) = 2^n, T(n,k) = sum_{j=0..k} T(n-1,j) for k=0..n-1.

%I A201634 #12 Mar 07 2020 12:07:58
%S A201634 1,1,2,1,3,4,1,4,8,8,1,5,13,21,16,1,6,19,40,56,32,1,7,26,66,122,154,
%T A201634 64,1,8,34,100,222,376,440,128,1,9,43,143,365,741,1181,1309,256,1,10,
%U A201634 53,196,561,1302,2483,3792,4048,512,1,11,64,260,821,2123,4606
%N A201634 Triangle read by rows, n>=0, k>=0, T(n,n) = 2^n, T(n,k) = sum_{j=0..k} T(n-1,j) for k=0..n-1.
%C A201634 Row sums are A014318.
%H A201634 Reinhard Zumkeller, <a href="/A201634/b201634.txt">Rows n = 0..125 of triangle, flattened</a>
%e A201634 [0] [1]
%e A201634 [1] [1, 2]
%e A201634 [2] [1, 3, 4]
%e A201634 [3] [1, 4, 8, 8]
%e A201634 [4] [1, 5, 13, 21, 16]
%e A201634 [5] [1, 6, 19, 40, 56, 32]
%e A201634 [6] [1, 7, 26, 66, 122, 154, 64]
%e A201634 [7] [1, 8, 34, 100, 222, 376, 440, 128]
%o A201634 (Sage)
%o A201634 @CachedFunction
%o A201634 def A201634(n, k):
%o A201634     if k==n: return 2^n
%o A201634     return add(A201634(n-1, j) for j in (0..k))
%o A201634 for n in (0..7) : print([A201634(n, k) for k in (0..n)])
%o A201634 (Haskell)
%o A201634 a201634 n k = a201634_tabl !! n !! k
%o A201634 a201634_row n = a201634_tabl !! n
%o A201634 a201634_tabl = iterate (\xs -> scanl1 (+) xs ++ [2 * last xs]) [1]
%o A201634 -- _Reinhard Zumkeller_, Sep 20 2014
%Y A201634 Cf. A247023 (mirrored).
%K A201634 nonn,tabl
%O A201634 0,3
%A A201634 _Peter Luschny_, Nov 14 2012