preserver_image / derivative_polynomial, shift 0

Image under a real-rootedness preserver. All operations

410 sequences

• basis-map coefficients of A021012 → D+1
  1 · -1, -1 · 0, 4, 2 · 0, 0, -18, -6 … row sums 1, -2, 6, -24, 120, -720, 5040, -40320, …
  (−∞, 0] rational roots
• A081578 → D+1
  1 · 2, 1 · 6, 7, 1 · 10, 27, 12, 1 … row sums 1, 3, 14, 50, 183, 637, 2188, 7380, …
  (−∞, 0] nonnegative
• A039991 → D
  0, -2 · 0, -6 · 0, -16, 0, 4 · 0, -40, 0, 20 … row sums 0, 0, -2, -6, -12, -20, -30, -42, …
  (−∞, ∞)
• A122076 → D+1
  2 · 5, 2 · 15, 12, 2 · 48, 60, 21, 2 … row sums 2, 7, 29, 131, 601, 2735, 12293, 54619, …
  (−∞, ∞) nonnegative
• basis-map coefficients of A145596 → D+1
  1 · 2, 2 · 0, 8, 3 · -8, 8, 20, 4 … row sums 1, 4, 11, 24, 45, 76, 119, 176, …
  (−∞, ∞)
• A178821 → D+1
  1 · 10, 5 · 45, 60, 15 · 140, 315, 210, 35 … row sums 1, 15, 120, 700, 3360, 14112, 53760, …
  (−∞, 0] nonnegative rational roots
• A052553 → D
  1 · 2 · 3, 2 · 4, 6 … row sums 0, 0, 1, 2, 5, 10, 20, 38, …
  (−∞, 0] nonnegative
• A120909 → D
  6 · 12, 24 · 18, 72, 72 · 24, 144, 288, 192 … row sums 0, 6, 36, 162, 648, 2430, 8748, 30618, …
  (−∞, 0] nonnegative rational roots
• A188440 → D+1
  1 · 1 · 3, 2 · 3, 2 … row sums 1, 1, 5, 5, 21, 21, 81, 81, …
  (−∞, 0] nonnegative rational roots
• narayana → D+1
  1 · 2, 1 · 4, 5, 1 · 7, 18, 9, 1 … row sums 1, 3, 10, 35, 126, 462, 1716, 6435, …
  (−∞, 0] nonnegative
• A173018 → D
  1 · 4, 2 · 11, 22, 3 · 26, 132, 78, 4 … row sums 0, 0, 1, 6, 36, 240, 1800, 15120, …
  (−∞, 0] nonnegative
• A055356 → D
  1 · 4, 4 · 11, 36, 18 · 26, 196, 288, 96 … row sums 0, 0, 1, 8, 65, 606, 6527, 80364, …
  (−∞, 0] nonnegative
• type_b_eulerian → γ (scale 1/4) → D+1
  1 · 1 · 2, 1 · 6, 5 … row sums 1, 1, 3, 11, 52, 300, 2040, 15960, …
  (−∞, 0] nonnegative
• A046716 → D
  1 · 3, 2 · 6, 16, 3 · 10, 58, 72, 4 … row sums 0, 1, 5, 25, 144, 961, 7313, 62525, …
  (−∞, 0] nonnegative
• A249248 → D
  3 · 7, 6 · 12, 44, 9 · 18, 138, 120, 12 … row sums 0, 3, 13, 65, 288, 1325, 6073, 28563, …
  (−∞, 0] nonnegative
• A334218 → D+1
  1 · 1 · 2, 2 · 3, 9, 3 … row sums 1, 1, 4, 15, 72, 420, 2880, 22680, …
  (−∞, 0] nonnegative
• A039991 → D+1
  1 · 1 · 2, -2, -1 · 4, -6, -3 … row sums 1, 1, -1, -5, -11, -19, -29, -41, …
  (−∞, ∞)
• A008277 → θ+1 → D
  2 · 6, 6 · 14, 36, 12 · 30, 150, 120, 20 … row sums 0, 2, 12, 62, 320, 1712, 9604, 56674, …
  (−∞, 0] nonnegative
• A132393 → D
  1 · 1, 2 · 2, 6, 3 · 6, 22, 18, 4 … row sums 0, 1, 3, 11, 50, 274, 1764, 13068, …
  (−∞, 0] nonnegative
• A172431 → ⊙ C(n,k+1) (shift -1) → D
  1 · 4, 2 · 9, 20, 3 · 16, 84, 60, 4 … row sums 0, 1, 6, 32, 164, 830, 4184, 21077, …
  (−∞, 0] nonnegative
• A008278 → x ↦ x+1 (shift +1) → D
  1 · 5, 2 · 23, 20, 3 · 109, 152, 57, 4 … row sums 0, 1, 7, 46, 322, 2427, 19645, 170236, …
  (−∞, 0] nonnegative
• A053123 → D+1
  1 · -1, -2 · -3, 2, 3 · -5, 14, -2, -4 … row sums 1, -3, 2, 3, -7, 4, 5, -11, …
  (−∞, ∞)
• A055356 → D+1
  1 · 1 · 2, 1 · 5, 8, 2 … row sums 1, 1, 3, 15, 101, 851, 8603, 101423, …
  (−∞, 0] nonnegative
• A152842 → D
  3 · 4, 6 · 7, 30, 27 · 8, 44, 72, 36 … row sums 0, 3, 10, 64, 160, 832, 1920, 9216, …
  (−∞, 0] nonnegative rational roots
• A011973 → x ↦ x−1/4 → D
  4 · 16 · 40, 32 · 80, 192 … row sums 0, 0, 4, 16, 72, 272, 1004, 3552, …
  (−∞, 0] nonnegative
• A033282 → D
  2 · 5, 10 · 9, 42, 42 · 14, 112, 252, 168 … row sums 0, 2, 15, 93, 546, 3140, 17901, 101675, …
  (−∞, 0] nonnegative
• basis-map coefficients of A132812 → D+1
  1 · 2, 2 · 0, 9, 3 · -12, 12, 24, 4 … row sums 1, 4, 12, 28, 55, 96, 154, 232, …
  (−∞, ∞)
• A154986 → D
  1 · 4, 2 · 11, 22, 3 · 24, 140, 72, 4 … row sums 0, 1, 6, 36, 240, 1800, 15120, 141120, …
  (−∞, 0] nonnegative
• A100862 → D
  1 · 3, 2 · 6, 12, 3 · 10, 42, 30, 4 … row sums 0, 1, 5, 21, 86, 355, 1497, 6475, …
  (−∞, 0] nonnegative
• A227450 → D
  4 · 12, 16 · 28, 96, 48 · 60, 400, 480, 128 … row sums 0, 4, 28, 172, 1068, 6924, 47276, …
  (−∞, 0] nonnegative
• A304236 → D
  3 · 6 · 9, 18 · 12, 54 … row sums 0, 0, 3, 6, 27, 66, 204, 522, …
  (−∞, 0] nonnegative
• A126351 → D+1
  1 · 3, 2 · 6, 13, 4 · 10, 47, 43, 8 … row sums 1, 5, 23, 108, 534, 2803, 15623, 92249, …
  (−∞, 0] nonnegative
• A132393 → D+1
  1 · 1, 1 · 1, 3, 1 · 2, 8, 6, 1 … row sums 1, 2, 5, 17, 74, 394, 2484, 18108, …
  (−∞, 0] nonnegative
• A008292 → γ (shift +1) → D+1
  1 · 1 · 3, 2 · 9, 8 … row sums 1, 1, 5, 17, 93, 513, 3477, 25569, …
  (−∞, 0] nonnegative
• A137320 → D
  2 · 2, 8 · 4, 24, 24 · 12, 88, 144, 64 … row sums 0, 2, 10, 52, 308, 2088, 16056, 138528, …
  (−∞, 0] nonnegative
• A141611: Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k).
  A007318 → −D²+(−1+n)D+1+n
  1 · 2, 2 · 3, 8, 3 · 4, 18, 18, 4 … row sums 1, 4, 14, 44, 128, 352, 928, 2368, …
  A141611 (−∞, 0] nonnegative
• A059473 → D+1
  1 · 4, 2 · 14, 18, 4 · 40, 96, 56, 8 … row sums 1, 6, 36, 200, 1068, 5544, 28192, …
  (−∞, 0] nonnegative
• A001263 → D+1
  1 · 2, 1 · 4, 5, 1 · 7, 18, 9, 1 … row sums 1, 3, 10, 35, 126, 462, 1716, 6435, …
  (−∞, 0] nonnegative
• A128099 → D+1
  1 · 1 · 3, 2 · 5, 4 … row sums 1, 1, 5, 9, 25, 53, 125, 273, …
  (−∞, 0] nonnegative
• A008292 → γ (scale 1/4) → D+1
  1 · 2 · 6, 2 · 24, 16 … row sums 1, 2, 8, 40, 240, 1680, 13440, 120960, …
  (−∞, 0] nonnegative
• A053117 → D+1
  1 · 2, 2 · -1, 8, 4 · -4, -4, 24, 8 … row sums 1, 4, 11, 24, 45, 76, 119, 176, …
  (−∞, ∞)
• A010054 → reverse → D
  3 · 0, 12 · 0, 0, 30 · 0, 0, 0, 60 … row sums 0, 3, 12, 30, 60, 105, 168, 252, …
  (−∞, 0] nonnegative rational roots
• A106246: Number triangle T(n,k)=C(n,k)C(2,n-k).
  A023531 → (1/2)D²+2D+1
  1 · 2, 1 · 1, 4, 1 · 0, 3, 6, 1 … row sums 1, 3, 6, 10, 15, 21, 28, 36, …
  A106246 (−∞, 0] nonnegative
• A039755 → D
  1 · 4, 2 · 13, 18, 3 · 40, 116, 48, 4 … row sums 0, 1, 6, 34, 208, 1396, 10232, 81320, …
  (−∞, 0] nonnegative
• A007318 → 2θ+1 → D
  3 · 6, 10 · 9, 30, 21 · 12, 60, 84, 36 … row sums 0, 3, 16, 60, 192, 560, 1536, 4032, …
  (−∞, 0] nonnegative rational roots
• A249250 → D+1
  1 · 4, 3 · 7, 12, 3 · 10, 39, 24, 3 … row sums 1, 7, 22, 76, 244, 772, 2392, 7312, …
  (−∞, 0] nonnegative
• A011973 → basis A007318 → D+1
  1 · 1 · 3, 1 · 5, 2 … row sums 1, 1, 4, 7, 18, 37, 84, 179, …
  (−∞, 0] nonnegative
• A010054 → D∘x → D
  (−∞, 0] nonnegative rational roots
• A078812 → D
  1 · 4, 2 · 10, 12, 3 · 20, 42, 24, 4 … row sums 0, 1, 6, 25, 90, 300, 954, 2939, …
  (−∞, 0] nonnegative
• A132440: Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the lis…
  A023531 → D
  1 · 0, 2 · 0, 0, 3 · 0, 0, 0, 4 … row sums 0, 1, 2, 3, 4, 5, 6, 7, …
  A132440 (−∞, 0] nonnegative rational roots