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A224791 Triangle T(n,k) read by rows: left edge is 0, 1, 2, ... (cf. A001477); otherwise each entry is sum of entry to left and entries immediately above it to left and right, with 1 for the missing right term at right edge.

%I A224791 #11 Nov 13 2019 01:48:26
%S A224791 0,1,2,2,5,8,3,10,23,32,4,17,50,105,138,5,26,93,248,491,630,6,37,156,
%T A224791 497,1236,2357,2988,7,50,243,896,2629,6222,11567,14556,8,65,358,1497,
%U A224791 5022,13873,31662,57785,72342,9,82,505,2360,8879,27774,73309,162756
%N A224791 Triangle T(n,k) read by rows: left edge is 0, 1, 2, ... (cf. A001477); otherwise each entry is sum of entry to left and entries immediately above it to left and right, with 1 for the missing right term at right edge.
%H A224791 Reinhard Zumkeller, <a href="/A224791/b224791.txt">Rows n = 0..120 of triangle, flattened</a>
%H A224791 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A224791 T(n,0) = n, T(n+1,k) = T(n+1,k-1) + T(n,k-1) + T(n,k) (0 < k <= n) and T(n+1,n+1) = T(n+1,n) + T(n,n) + 1.
%e A224791 Triangle begins:
%e A224791   0;
%e A224791   1,  2;
%e A224791   2,  5,  8;
%e A224791   3, 10, 23,  32;
%e A224791   4, 17, 50, 105, 138;
%p A224791 T:= proc(n, k) option remember;
%p A224791       if k=0 then n
%p A224791     elif k=n then T(n,n-1) + T(n-1,n-1) + 1
%p A224791     else T(n,k-1) + T(n-1,k-1) + T(n-1, k)
%p A224791       fi
%p A224791     end:
%p A224791 seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Nov 12 2019
%t A224791 T[n_, k_]:= T[n, k]= If[k==0, n, If[k==n , T[n, n-1] + T[n-1, n-1] + 1, T[n, k-1] + T[n-1, k-1] + T[n-1, k]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 12 2019 *)
%o A224791 (Haskell)
%o A224791 a224791 n k = a224791_tabl !! n !! k
%o A224791 a224791_row n = a224791_tabl !! n
%o A224791 a224791_tabl = iterate
%o A224791    (\row -> scanl1 (+) $ zipWith (+) ([1] ++ row) (row ++ [1])) [0]
%o A224791 (PARI) T(n,k) = if(k==0, n, if(k==n, T(n,n-1) + T(n-1,n-1) + 1, T(n,k-1) + T(n-1,k-1) + T(n-1, k) )); \\ _G. C. Greubel_, Nov 12 2019
%o A224791 (Sage)
%o A224791 @CachedFunction
%o A224791 def T(n, k):
%o A224791     if (k==0): return n
%o A224791     elif (k==n): return T(n,n-1) + T(n-1,n-1) + 1
%o A224791     else: return T(n,k-1) + T(n-1,k-1) + T(n-1, k)
%o A224791 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 12 2019
%Y A224791 Cf. A051601, A059283.
%K A224791 nonn,tabl
%O A224791 0,3
%A A224791 _Reinhard Zumkeller_, Apr 18 2013