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A225413 Triangle read by rows: T(n,k) = (A101164(n,k) - A014473(n,k))/2.

%I A225413 #18 Apr 09 2024 05:37:46
%S A225413 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,3,0,0,0,0,6,12,6,0,0,0,0,10,30,
%T A225413 30,10,0,0,0,0,15,60,91,60,15,0,0,0,0,21,105,215,215,105,21,0,0,0,0,
%U A225413 28,168,435,590,435,168,28,0,0,0,0,36,252,791,1365,1365,791,252,36,0,0
%N A225413 Triangle read by rows: T(n,k) = (A101164(n,k) - A014473(n,k))/2.
%C A225413 Has opposite parity to A140356, A155454.
%H A225413 Reinhard Zumkeller, <a href="/A225413/b225413.txt">Rows n = 0..100 of table, flattened</a>
%F A225413 T(n, k) = (A101164(n,k) - A014473(n,k))/2.
%F A225413 T(n, k) = (A008288(n,k) - 2*A007318(n,k) + 1)/2.
%F A225413 From _G. C. Greubel_, Apr 08 2024: (Start)
%F A225413 T(n, n-k) = T(n, k).
%F A225413 Sum_{k=0..n} T(n, k) = (A000129(n+1) + n + 1 - 2^(n+1))/2.
%F A225413 Sum_{k=0..n} (-1)^k*T(n, k) = A121262(n) - [n=0]. (End)
%e A225413 Triangle begins as:
%e A225413   0;
%e A225413   0,  0;
%e A225413   0,  0,  0;
%e A225413   0,  0,  0,   0;
%e A225413   0,  0,  1,   0,    0;
%e A225413   0,  0,  3,   3,    0,    0;
%e A225413   0,  0,  6,  12,    6,    0,    0;
%e A225413   0,  0, 10,  30,   30,   10,    0,    0;
%e A225413   0,  0, 15,  60,   91,   60,   15,    0,    0;
%e A225413   0,  0, 21, 105,  215,  215,  105,   21,    0,    0;
%e A225413   0,  0, 28, 168,  435,  590,  435,  168,   28,    0,   0;
%e A225413   0,  0, 36, 252,  791, 1365, 1365,  791,  252,   36,   0,  0;
%e A225413   0,  0, 45, 360, 1330, 2800, 3571, 2800, 1330,  360,  45,  0,  0;
%e A225413   0,  0, 55, 495, 2106, 5250, 8197, 8197, 5250, 2106, 495, 55,  0,  0;
%t A225413 T[n_, k_]:= ((-1)^(n-k)*Hypergeometric2F1[-n+k,k+1,1,2] - 2*Binomial[n, k] +1)/2;
%t A225413 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 08 2024 *)
%o A225413 (Haskell)
%o A225413 a225413 n k = a225413_tabl !! n !! k
%o A225413 a225413_row n = a225413_tabl !! n
%o A225413 a225413_tabl = map (map (`div` 2)) $
%o A225413                zipWith (zipWith (-)) a101164_tabl a014473_tabl
%o A225413 -- _Reinhard Zumkeller_, Jul 30 2013
%o A225413 (Magma)
%o A225413 A008288:= func< n,k | (&+[Binomial(n-j, j)*Binomial(n-2*j, k-j): j in [0..k]]) >;
%o A225413 A225413:= func< n,k | (A008288(n,k) - 2*Binomial(n,k) + 1)/2 >;
%o A225413 [A225413(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 08 2024
%o A225413 (SageMath)
%o A225413 def A008288(n,k): return sum(binomial(n-j,j)*binomial(n-2*j,k-j) for j in range(k+1))
%o A225413 def A225413(n,k): return (A008288(n,k) -2*binomial(n,k) +1)//2
%o A225413 flatten([[A225413(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Apr 08 2024
%Y A225413 Cf. A000129, A007318, A008288, A014473, A101164, A121262, A140356, A155454.
%Y A225413 3rd column = A000217 (triangular numbers).
%Y A225413 4th column = A027480 (n(n+1)(n+2)/2).
%K A225413 nonn,easy,tabl
%O A225413 0,18
%A A225413 _Jeremy Gardiner_, Jul 28 2013