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A201635 Triangle formed by T(n,n) = (-1)^n*Sum_{j=0..n} C(-n,j), T(n,k) = Sum_{j=0..k} T(n-1,j) for k=0..n-1, and n>=0, read by rows.

%I A201635 #10 Mar 01 2019 08:04:19
%S A201635 1,1,0,1,1,2,1,2,4,6,1,3,7,13,22,1,4,11,24,46,80,1,5,16,40,86,166,296,
%T A201635 1,6,22,62,148,314,610,1106,1,7,29,91,239,553,1163,2269,4166,1,8,37,
%U A201635 128,367,920,2083,4352,8518,15792,1,9,46,174,541,1461,3544,7896
%N A201635 Triangle formed by T(n,n) = (-1)^n*Sum_{j=0..n} C(-n,j), T(n,k) = Sum_{j=0..k} T(n-1,j) for k=0..n-1, and n>=0, read by rows.
%C A201635 Notation: If a sequence id is starred then the offset and/or some terms are different. Starred terms indicate the variance.
%C A201635 Row sums: [A026641 ] [1,  1,  4,  13,  46,  166,   610]
%C A201635 --
%C A201635 T(j+2, 2) [A000124*] [1*, 2 , 4,   7,  11,  16,     22]
%C A201635 T(j+3, 3) [A003600*] [1*, 2*, 6,  13,  24,  40,     62]
%C A201635 --
%C A201635 T(j  , j) [A072547 ] [1,  0,  2,   6,  22,  80,    296]
%C A201635 T(j+1, j) [A026641 ] [1,  1,  4,  13,  46,  166,   610]
%C A201635 T(j+2, j) [A014300 ] [1,  2,  7,  24,  86,  314,  1163]
%C A201635 T(j+3, j) [A014301*] [1,  3, 11,  40, 148,  553,  2083]
%C A201635 T(j+4, j) [A172025 ] [1,  4, 16,  62, 239,  920,  3544]
%C A201635 T(j+5, j) [A172061 ] [1,  5, 22,  91, 367, 1461,  5776]
%C A201635 T(j+6, j) [A172062 ] [1,  6, 29, 128, 541, 2232,  9076]
%C A201635 T(j+7, j) [A172063 ] [1,  7, 37, 174, 771, 3300, 13820]
%C A201635 --
%C A201635 T(2j  ,j) [Central ] [1,  1,  7,  40, 239, 1461,  9076]
%C A201635 T(2j+1,j) [A183160 ] [1,  2, 11,  62, 367, 2232, 13820]
%C A201635 T(2j+2,j) [        ] [1,  3, 16,  91, 541, 3300, 20476]
%C A201635 T(2j+3,j) [A199033*] [1,  4, 22, 128, 771, 4744, 29618]
%H A201635 G. C. Greubel, <a href="/A201635/b201635.txt">Rows n=0..100 of triangle, flattened</a>
%e A201635 Triangle begins as:
%e A201635 [n]|k->
%e A201635 [0] 1
%e A201635 [1] 1, 0
%e A201635 [2] 1, 1,  2
%e A201635 [3] 1, 2,  4,  6
%e A201635 [4] 1, 3,  7, 13,  22
%e A201635 [5] 1, 4, 11, 24,  46,  80
%e A201635 [6] 1, 5, 16, 40,  86, 166, 296
%e A201635 [7] 1, 6, 22, 62, 148, 314, 610, 1106.
%p A201635 A201635 := proc(n,k) option remember; local j;
%p A201635 if n=k then (-1)^n*add(binomial(-n,j), j=0..n)
%p A201635 else add(A201635(n-1,j), j=0..k) fi end:
%p A201635 for n from 0 to 7 do seq(A(n,k), k=0..n) od;
%t A201635 T[n_, k_]:= T[n, k]= If[k==n, (-1)^n*Sum[Binomial[-n, j], {j, 0, n}], Sum[T[n-1, j], {j, 0, k}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 27 2019 *)
%o A201635 (Sage)
%o A201635 @CachedFunction
%o A201635 def A201635(n, k):
%o A201635     if n==k: return (-1)^n*add(binomial(-n, j) for j in (0..n))
%o A201635     return add(A201635(n-1, j) for j in (0..k))
%o A201635 for n in (0..7) : [A201635(n, k) for k in (0..n)]
%o A201635 (PARI)
%o A201635 {T(n,k) = if(k==n, (-1)^n*sum(j=0,n, binomial(-n,j)), sum(j=0,k, T(n-1,j)))};
%o A201635 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Feb 27 2019
%K A201635 nonn,tabl
%O A201635 0,6
%A A201635 _Peter Luschny_, Nov 14 2012