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A051666 Rows of triangle formed using Pascal's rule except begin and end n-th row with n^2.

%I A051666 #26 Jul 02 2025 16:01:58
%S A051666 0,1,1,4,2,4,9,6,6,9,16,15,12,15,16,25,31,27,27,31,25,36,56,58,54,58,
%T A051666 56,36,49,92,114,112,112,114,92,49,64,141,206,226,224,226,206,141,64,
%U A051666 81,205,347,432,450,450,432,347,205,81,100,286,552,779,882,900,882,779
%N A051666 Rows of triangle formed using Pascal's rule except begin and end n-th row with n^2.
%C A051666 Row sums give 6*2^n - 4*n - 6 (A051667).
%C A051666 Central terms: T(2*n,n) = 2 * A220101(n). - _Reinhard Zumkeller_, Aug 05 2013
%C A051666 For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - _Boris Putievskiy_, Aug 19 2013
%C A051666 For a closed-form formula for generalized Pascal's triangle see A228576. - _Boris Putievskiy_, Sep 09 2013
%H A051666 Reinhard Zumkeller, <a href="/A051666/b051666.txt">Rows n = 0..100 of table, flattened</a>
%e A051666 Triangle begins:
%e A051666    0;
%e A051666    1,  1;
%e A051666    4,  2,  4;
%e A051666    9,  6,  6,  9;
%e A051666   16, 15, 12, 15, 16;
%e A051666   ...
%t A051666 T[n_, 0] := n^2; T[n_, n_] := n^2;
%t A051666 T[n_, k_] := T[n, k] = T[n-1, k-1] + T[n-1, k];
%t A051666 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 13 2018 *)
%o A051666 (Haskell)
%o A051666 a051666 n k = a051666_tabl !! n !! k
%o A051666 a051666_row n = a051666_tabl !! n
%o A051666 a051666_tabl = map fst $ iterate
%o A051666    (\(vs, w:ws) -> (zipWith (+) ([w] ++ vs) (vs ++ [w]), ws))
%o A051666    ([0], [1, 3 ..])
%o A051666 -- _Reinhard Zumkeller_, Aug 05 2013
%Y A051666 Cf. A007318, A000290, A005408, A228196, A228576.
%K A051666 easy,nonn,tabl
%O A051666 0,4
%A A051666 _Asher Auel_
%E A051666 More terms from _James Sellers_