preserver_image / reversal, shift 0

Image under a real-rootedness preserver. All operations

164 sequences

• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
  1 · 3, 2 · 9, 12, 4 · 27, 54, 36, 8 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A038220 (−∞, −1] nonnegative rational roots
• A056939: Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.
  A108838 → ⊙ C(n+1,k+1) (shift +1)
  1 · 1, 1 · 1, 4, 1 · 1, 10, 10, 1 … row sums 1, 2, 6, 22, 92, 422, 2074, 10754, …
  A056939 (−∞, 0] nonnegative
• A099093 → reverse
  1 · 3 · 9, 3 · 27, 18 … row sums 1, 3, 12, 45, 171, 648, 2457, 9315, …
  (−∞, 0] nonnegative
• A143495 → reverse
  1 · 1, 3 · 1, 7, 9 · 1, 12, 37, 27 … row sums 1, 4, 17, 77, 372, 1915, 10481, 60814, …
  (−∞, 0] nonnegative
• A100862: Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n…
  1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 … row sums 1, 2, 5, 14, 43, 142, 499, 1850, …
  A100862 (−∞, 0) nonnegative
• A028326: Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).
  2 · 2, 2 · 2, 4, 2 · 2, 6, 6, 2 … row sums 2, 4, 8, 16, 32, 64, 128, 256, …
  A028326 (−∞, 0) nonnegative rational roots
• A154986: Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.
  1 · 1, 1 · 1, 4, 1 · 1, 11, 11, 1 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  A154986 (−∞, 0) nonnegative
• A034870: Even-numbered rows of Pascal's triangle.
  1 · 1, 2, 1 · 1, 4, 6, 4, 1 · 1, 6, 15, 20, 15, 6, 1 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A034870 (−∞, −1] nonnegative rational roots
• A102547: Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318).
  1 · 1 · 1 · 1, 1 … row sums 1, 1, 1, 2, 3, 4, 6, 9, …
  A102547 (−∞, 0] nonnegative
• A145596: basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x -> x + 1
  1 · 2, 2 · 3, 8, 3 · 4, 20, 20, 4 … row sums 1, 4, 14, 48, 165, 572, 2002, 7072, …
  A145596 (−∞, 0) nonnegative
• A064861 → reverse
  2, 3, 1 · 4, 8, 5, 1 · 4, 12, 13, 6, 1 … row sums 6, 18, 36, 108, 216, 648, 1296, 3888, …
  (−∞, 0] nonnegative rational roots
• A113413 → reverse
  1 · 1, 2 · 1, 4, 2 · 1, 6, 8, 2 … row sums 1, 3, 7, 17, 41, 99, 239, 577, …
  (−∞, 0] nonnegative
• A007318 → (θ+1)² → reverse
  1 · 4, 1 · 9, 8, 1 · 16, 27, 12, 1 … row sums 1, 5, 18, 56, 160, 432, 1120, 2816, …
  (−∞, 0] nonnegative
• A114687 → reverse
  1 · 2, 1 · 4, 6, 1 · 8, 24, 12, 1 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  (−∞, 0] nonnegative
• A122431 → reverse
  1, 3, 3, 1 · 1, 3, 3, 1 · 1, 3, 3, 1 · 1, 3, 3, 1 … row sums 8, 8, 8, 8, 8, 8, 8, 8, …
  (−∞, 0] nonnegative rational roots
• A059473: Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...
  1 · 2, 2 · 4, 10, 4 · 8, 32, 32, 8 … row sums 1, 4, 18, 80, 356, 1584, 7048, 31360, …
  A059473 (−∞, 0) nonnegative
• A143683: Pascal-(1,8,1) array.
  1 · 1, 1 · 1, 10, 1 · 1, 19, 19, 1 … row sums 1, 2, 12, 40, 176, 672, 2752, 10880, …
  A143683 (−∞, 0) nonnegative
• A007318 → θ+1+n → reverse
  1 · 3, 2 · 5, 8, 3 · 7, 18, 15, 4 … row sums 1, 5, 16, 44, 112, 272, 640, 1472, …
  (−∞, 0] nonnegative rational roots
• A081582: Pascal-(1,7,1) array.
  1 · 1, 1 · 1, 9, 1 · 1, 17, 17, 1 … row sums 1, 2, 11, 36, 149, 550, 2143, 8136, …
  A081582 (−∞, 0) nonnegative
• A081581: Pascal-(1,6,1) array.
  1 · 1, 1 · 1, 8, 1 · 1, 15, 15, 1 … row sums 1, 2, 10, 32, 124, 440, 1624, 5888, …
  A081581 (−∞, 0) nonnegative
• A091044: One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
  1 · 2, 2 · 3, 10, 3 · 4, 28, 28, 4 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A091044 (−∞, 0) nonnegative
• A102662 → reverse
  1 · 3, 1 · 3, 5, 1 · 3, 11, 7, 1 … row sums 1, 4, 9, 22, 53, 128, 309, 746, …
  (−∞, 0] nonnegative
• narayana
  1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 … row sums 1, 2, 5, 14, 42, 132, 429, 1430, …
  (−∞, 0) nonnegative
• A007318 → (1/2+(1/2)n)θ+1+n → reverse
  1 · 3, 2 · 6, 9, 3 · 10, 24, 18, 4 … row sums 1, 5, 18, 56, 160, 432, 1120, 2816, …
  (−∞, 0] nonnegative rational roots
• A007318 → (1/3+(1/2)n+(1/6)n²)θ+1+(3/2)n+(1/2)n² → reverse
  1 · 4, 3 · 10, 16, 6 · 20, 50, 40, 10 … row sums 1, 7, 32, 120, 400, 1232, 3584, 9984, …
  (−∞, 0] nonnegative rational roots
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
  1 · 1, 4 · 1, 8, 16 · 1, 12, 48, 64 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A013611 (−∞, 0) [−1, 0] nonnegative rational roots
• A007318 → basis A049020 → reverse
  1 · 1, 2 · 1, 5, 5 · 1, 9, 22, 15 … row sums 1, 3, 11, 47, 227, 1215, 7107, 44959, …
  (−∞, 0] nonnegative
• A340556: E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.
  A163936 → reverse
  1 · 0, 1 · 0, 1, 2 · 0, 1, 8, 6 … row sums 1, 1, 3, 15, 105, 945, 10395, 135135, …
  A340556 (−∞, 0] nonnegative
• A063967 → reverse
  1 · 1, 1 · 1, 3, 2 · 1, 5, 7, 3 … row sums 1, 2, 6, 16, 44, 120, 328, 896, …
  (−∞, 0] nonnegative
• A216973: Exponential Riordan array [x*exp(x),x].
  A007318 → D
  1 · 2, 2 · 3, 6, 3 · 4, 12, 12, 4 … row sums 0, 1, 4, 12, 32, 80, 192, 448, …
  A216973 (−∞, −1] nonnegative rational roots
• A094816: Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.
  1 · 1, 1 · 1, 3, 1 · 1, 8, 6, 1 … row sums 1, 2, 5, 16, 65, 326, 1957, 13700, …
  A094816 (−∞, 0) nonnegative
• A023531 → 2D+1+2n → reverse
  1 · 3, 2 · 5, 4 · 7, 6 … row sums 1, 5, 9, 13, 17, 21, 25, 29, …
  (−∞, 0] nonnegative rational roots
• A110441 → reverse
  1 · 1, 3 · 1, 6, 7 · 1, 9, 23, 15 … row sums 1, 4, 14, 48, 164, 560, 1912, 6528, …
  (−∞, 0] nonnegative
• A249248 → reverse
  1 · 3, 1 · 3, 7, 1 · 3, 22, 12, 1 … row sums 1, 4, 11, 38, 131, 490, 1897, 7714, …
  (−∞, 0] nonnegative
• A135065: A127733 * A007318 as infinite lower triangular matrices.
  A007318 → ×(1+2n+n²)
  1 · 4, 4 · 9, 18, 9 · 16, 48, 48, 16 … row sums 1, 8, 36, 128, 400, 1152, 3136, 8192, …
  A135065 (−∞, −1] nonnegative rational roots
• A010054: a(n) = 1 if n is a triangular number, otherwise 0.
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A010054 (−∞, −1] [−1, 0] [−3−2√2, −3+2√2] nonnegative rational roots
• A391533 → reverse
  1, 4, 4 · 1, 5, 8, 4 · 1, 6, 13, 12, 4 … row sums 9, 18, 36, 72, 144, 288, 576, 1152, …
  (−∞, 0] nonnegative rational roots
• A172431 → ⊙ C(n,k+1) (shift -1) → reverse
  1 · 1, 1 · 1, 4, 1 · 1, 10, 9, 1 … row sums 1, 2, 6, 21, 80, 322, 1347, 5798, …
  (−∞, 0] nonnegative
• A008277: Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
  1 · 1, 1 · 1, 3, 1 · 1, 7, 6, 1 … row sums 1, 2, 5, 15, 52, 203, 877, 4140, …
  A008277 (−∞, 0) nonnegative
• A249250 → reverse
  1 · 3, 1 · 3, 6, 1 · 3, 15, 9, 1 … row sums 1, 4, 10, 28, 76, 208, 568, 1552, …
  (−∞, 0] nonnegative
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
  1 · 4, 1 · 16, 8, 1 · 64, 48, 12, 1 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A038231 (−∞, −1] nonnegative rational roots
• A137320 → reverse
  1 · 2 · 4, 2 · 8, 12, 4 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  (−∞, 0] nonnegative rational roots
• A062196 → reverse
  1 · 3, 1 · 6, 8, 1 · 10, 30, 15, 1 … row sums 1, 4, 15, 56, 210, 792, 3003, 11440, …
  (−∞, 0] nonnegative
• A152198: Triangle read by rows, A007318 rows repeated.
  1 · 1 · 1, 1 · 1, 1 … row sums 1, 1, 2, 2, 4, 4, 8, 8, …
  A152198 A152815 (−∞, 0] nonnegative rational roots
• A114655 → reverse
  2 · 2, 4 · 2, 12, 8 · 2, 24, 48, 16 … row sums 2, 6, 22, 90, 394, 1806, 8558, 41586, …
  (−∞, 0] nonnegative
• A008278 → x ↦ x+1 (shift +1) → reverse
  1 · 1, 2 · 1, 5, 5 · 1, 10, 23, 15 … row sums 1, 3, 11, 49, 257, 1539, 10299, 75905, …
  (−∞, 0] nonnegative
• A099097 → reverse
  1 · 3 · 9, 1 · 27, 6 … row sums 1, 3, 10, 33, 109, 360, 1189, 3927, …
  (−∞, 0] nonnegative
• A130850: Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.
  A028246 → reverse
  1 · 1, 1 · 2, 3, 1 · 6, 12, 7, 1 … row sums 1, 2, 6, 26, 150, 1082, 9366, 94586, …
  A130850 (−∞, −1] nonnegative
• A063007: T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
  1 · 1, 2 · 1, 6, 6 · 1, 12, 30, 20 … row sums 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  A063007 (−1, 0) nonnegative
• A054142 → reverse
  1 · 1, 1 · 1, 3, 1 · 1, 6, 5, 1 … row sums 1, 2, 5, 13, 34, 89, 233, 610, …
  (−∞, 0] nonnegative