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 A134636 Triangle formed by Pascal's rule given borders = 2n+1. 7
 1, 3, 3, 5, 6, 5, 7, 11, 11, 7, 9, 18, 22, 18, 9, 11, 27, 40, 40, 27, 11, 13, 38, 67, 80, 67, 38, 13, 15, 51, 105, 147, 147, 105, 51, 15, 17, 66, 156, 252, 294, 252, 156, 66, 17, 19, 83, 222, 408, 546, 546, 408, 222, 83, 19, 21, 102, 305, 630, 954, 1092, 954, 630, 305, 102, 21 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Row sums = A048487: (1, 6, 16, 36, 76, 156, ...). LINKS Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened FORMULA Triangle, given borders = (1, 3, 5, 7, 9, ...); apply Pascal's rule T(n,k) = T(n-1,k) P T(n-1,k-1). T(n,k) = A051601(n,k) + A051597(n,k); T(n,k) mod 2 = A047999(n,k). - Reinhard Zumkeller, Nov 23 2012 Closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 19 2013 EXAMPLE First few rows of the triangle: 1; 3, 3; 5, 6, 5; 7, 11, 11, 7; 9, 18, 22, 18, 9; 11, 27, 40, 40, 27, 11; 13, 38, 67, 80, 67, 38, 13; ... MAPLE T:= proc(n, k) option remember; `if`(k<0 or k>n, 0, `if`(k=0 or k=n, 2*n+1, T(n-1, k-1) + T(n-1, k) )) end: seq(seq(T(n, k), k=0..n), n=0..14); # Alois P. Heinz, May 26 2013 MATHEMATICA NestList[Append[Prepend[Map[Apply[Plus, #] &, Partition[#, 2, 1]], #[[1]] + 2], #[[1]] + 2] &, {1}, 10] // Grid (* Geoffrey Critzer, May 26 2013 *) T[n_, k_] := Binomial[n, k-1] + Binomial[n, k] + 2 Binomial[n, k+1] + Binomial[n, n-k+1]; Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 07 2021 *) PROG (Haskell) a134636 n k = a134636_tabl !! n !! k a134636_row n = a134636_tabl !! n a134636_tabl = iterate (\row -> zipWith (+) ([2] ++ row) (row ++ [2])) [1] -- Reinhard Zumkeller, Nov 23 2012 CROSSREFS Cf. A007318, A048487, A051601, A051597. Sequence in context: A317745 A204250 A131948 * A088564 A161560 A078796 Adjacent sequences: A134633 A134634 A134635 * A134637 A134638 A134639 KEYWORD nonn,tabl AUTHOR Gary W. Adamson, Nov 04 2007 EXTENSIONS Offset changed by Reinhard Zumkeller, Nov 23 2012 STATUS approved

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Last modified February 2 08:56 EST 2023. Contains 360002 sequences. (Running on oeis4.)