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

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, 56, 36, 49, 92, 114, 112, 112, 114, 92, 49, 64, 141, 206, 226, 224, 226, 206, 141, 64, 81, 205, 347, 432, 450, 450, 432, 347, 205, 81, 100, 286, 552, 779, 882, 900, 882, 779

Offset

0,4

Comments

Row sums give 6*2^n - 4*n - 6 (A051667).

Central terms: T(2*n,n) = 2 * A220101(n). - Reinhard Zumkeller, Aug 05 2013

For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 19 2013

For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013

Links

Reinhard Zumkeller, Rows n = 0..100 of table, flattened

Example

Triangle begins:
   0;
   1,  1;
   4,  2,  4;
   9,  6,  6,  9;
  16, 15, 12, 15, 16;
  ...

Mathematica

T[n_, 0] := n^2; T[n_, n_] := n^2;
T[n_, k_] := T[n, k] = T[n-1, k-1] + T[n-1, k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 13 2018 *)

Prog

(Haskell)
a051666 n k = a051666_tabl !! n !! k
a051666_row n = a051666_tabl !! n
a051666_tabl = map fst $ iterate
   (\(vs, w:ws) -> (zipWith (+) ([w] ++ vs) (vs ++ [w]), ws))
   ([0], [1, 3 ..])
-- Reinhard Zumkeller, Aug 05 2013

Crossrefs

Cf. A007318, A000290, A005408, A228196, A228576.

Sequence in context: A128136 A048147 A203001 * A011382 A011302 A302603

Adjacent sequences: A051663 A051664 A051665 * A051667 A051668 A051669

Keyword

easy,nonn,tabl

Author

Asher Auel

Extensions

More terms from James A. Sellers

Status

approved