Proof methods / preserver_image

Image under a real-rootedness preserver

1,387 sequences

• A391533 → D
  4, 2 · 8, 10, 3 · 12, 26, 18, 4 · 16, 50, 57, 28, 5 … Σ 6, 21, 60, 156, 384, 912, 2112, 4800, …
  (−∞, 0]nonnegativerational roots
• A144438 → D+1
  1 · 2, 1 · 6, 7, 1 · 15, 42, 17, 1 … Σ 1, 3, 14, 75, 471, 3402, 27844, 254970, …
  (−∞, 0]nonnegative
• A136572 → θ+1
  1 · 0, 2 · 0, 0, 6 · 0, 0, 0, 24 … Σ 1, 2, 6, 24, 120, 720, 5040, 40320, …
  (−∞, 0]nonnegativerational roots
• A132440 → (θ+1)²
  1 · 0, 8 · 0, 0, 27 · 0, 0, 0, 64 … Σ 0, 1, 8, 27, 64, 125, 216, 343, …
  (−∞, 0]nonnegativerational roots
• A132774 → θ+1
  1 · 2, 6 · 0, 8, 15 · 0, 0, 18, 28 … Σ 1, 8, 23, 46, 77, 116, 163, 218, …
  (−∞, 0]nonnegativerational roots
• A126216 → D+1
  1 · 3, 1 · 10, 7, 1 · 35, 39, 12, 1 … Σ 1, 4, 18, 87, 439, 2278, 12052, 64669, …
  (−∞, 0]nonnegative
• A121574 → basis A007318
  1 · 3, 1 · 10, 7, 1 · 33, 35, 11, 1 … Σ 1, 4, 18, 80, 356, 1584, 7048, 31360, …
  (−∞, −1]nonnegative
• A122848 → θ+1
  1 · 0, 2 · 0, 2, 3 · 0, 0, 9, 4 … Σ 1, 2, 5, 13, 38, 116, 382, 1310, …
  (−∞, 0]nonnegative
• A113953: A Jacobsthal triangle.
  1 · 0, 1 · 0, 2, 1 · 0, 0, 4, 1 … Σ 1, 1, 3, 5, 11, 21, 43, 85, …
  (−8, 0]nonnegative
• A008292 → γ (scale 1/2) (shift +1) → θ+1
  1 · 1 · 1, 2 · 1, 8 … Σ 1, 1, 3, 9, 35, 155, 791, 4529, …
  (−∞, 0]nonnegative
• A106800 → D
  1 · 3, 2 · 6, 14, 3 · 10, 50, 45, 4 … Σ 0, 0, 1, 5, 23, 109, 544, 2876, …
  (−∞, 0]nonnegative
• A145597: Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.
  1 · 3, 3 · 6, 15, 6 · 10, 45, 45, 10 … Σ 1, 6, 27, 110, 429, 1638, 6188, 23256, …
  (−∞, 0)nonnegative
• A082985 → (θ+1)²
  1 · 1, 12 · 1, 20, 45 · 1, 28, 126, 112 … Σ 1, 13, 66, 267, 985, 3444, 11597, …
  (−∞, 0]nonnegative
• A347823 → (θ+1)²
  1 · 2, 12 · 3, 32, 45 · 4, 60, 162, 112 … Σ 1, 14, 80, 338, 1216, 3952, 11968, …
  (−∞, 0]nonnegative
• A131431 → D+1
  1 · 4, 4 · 0, 14, 7 · 0, 0, 30, 10 … Σ 1, 8, 21, 40, 65, 96, 133, 176, …
  (−∞, 0]nonnegativerational roots
• A062264 → (θ+1)²
  1 · 1, 20 · 1, 48, 135 · 1, 84, 567, 560 … Σ 1, 21, 184, 1212, 6975, 37147, 188188, …
  (−∞, 0]nonnegative
• A137597: Triangle read by rows: A008277 * A007318.
  1 · 2, 1 · 5, 5, 1 · 15, 22, 9, 1 … Σ 1, 3, 11, 47, 227, 1215, 7107, 44959, …
  (−∞, −1]nonnegative
• A132393 → θ+1
  1 · 0, 2 · 0, 2, 3 · 0, 4, 9, 4 … Σ 1, 2, 5, 17, 74, 394, 2484, 18108, …
  (−∞, 0]nonnegative
• A122542 → θ+1
  1 · 0, 2 · 0, 4, 3 · 0, 4, 12, 4 … Σ 1, 2, 7, 20, 57, 158, 431, 1160, …
  (−∞, 0]nonnegative
• A131431 → reverse
  1 · 4 · 7 · 10 … Σ 1, 4, 7, 10, 13, 16, 19, 22, …
  (−∞, 0]nonnegativerational roots
• A049020 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 7, 6 · 5, 25, 41, 22 … Σ 1, 3, 15, 93, 681, 5691, 53079, 544053, …
  [−1, 0]nonnegative
• A131689 → (θ+1)²
  1 · 0, 4 · 0, 4, 18 · 0, 4, 54, 96 … Σ 1, 4, 22, 154, 1306, 12994, 148282, …
  (−∞, 0]nonnegative
• A131689 → θ+1
  1 · 0, 2 · 0, 2, 6 · 0, 2, 18, 24 … Σ 1, 2, 8, 44, 308, 2612, 25988, 296564, …
  (−∞, 0]nonnegative
• A267482 → D
  1 · 1, 2 · -2, 2, 3 · -2, -6, 3, 4 … Σ 0, 1, 3, 3, -1, -6, -6, 1, …
  (−∞, ∞)
• A013611 → D+1
  1 · 5, 4 · 9, 40, 16 · 13, 108, 240, 64 … Σ 1, 9, 65, 425, 2625, 15625, 90625, …
  (−∞, 0]nonnegativerational roots
• A081578 → x ↦ x+1
  1 · 2, 1 · 7, 7, 1 · 20, 30, 12, 1 … Σ 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  (−∞, −1]nonnegative
• A085478 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 9, 20, 13 … Σ 1, 3, 11, 43, 171, 683, 2731, 10923, …
  [−1, 0)nonnegative
• A008459 → θ+1
  1 · 1, 2 · 1, 8, 3 · 1, 18, 27, 4 … Σ 1, 3, 12, 50, 210, 882, 3696, 15444, …
  (−∞, 0]nonnegative
• A081580 → D+1
  1 · 2, 1 · 8, 9, 1 · 14, 39, 16, 1 … Σ 1, 3, 18, 70, 303, 1197, 4756, 18396, …
  (−∞, 0]nonnegative
• A152842 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 6, 8 · 1, 10, 32, 32 … Σ 1, 5, 15, 75, 225, 1125, 3375, 16875, …
  [−1, 0]nonnegativerational roots
• A028326 → (θ+1)²
  2 · 2, 8 · 2, 16, 18 · 2, 24, 54, 32 … Σ 2, 10, 36, 112, 320, 864, 2240, 5632, …
  (−∞, 0]nonnegative
• A143495 → x ↦ x+1
  1 · 4, 1 · 17, 9, 1 · 77, 64, 15, 1 … Σ 1, 5, 27, 157, 979, 6517, 46107, …
  (−∞, −1]nonnegative
• A049020: Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.
  1 · 1, 1 · 2, 3, 1 · 5, 10, 6, 1 … Σ 1, 2, 6, 22, 94, 454, 2430, 14214, …
  (−∞, −1]nonnegative
• A086645 → x ↦ x+1
  1 · 2, 1 · 8, 8, 1 · 32, 48, 18, 1 … Σ 1, 3, 17, 99, 577, 3363, 19601, 114243, …
  (−∞, −1]nonnegative
• A116666 → reverse
  1 · 3, 1 · 5, 6, 1 · 7, 15, 9, 1 … Σ 1, 4, 12, 32, 80, 192, 448, 1024, …
  (−∞, 0]nonnegativerational roots
• A267482 → D+1
  1 · 2, 1 · 0, 3, 1 · -3, 0, 4, 1 … Σ 1, 3, 4, 2, -3, -7, -5, 3, …
  (−∞, ∞)
• basis-map coefficients of A030528, shifted → θ+1
  1 · 0, -2 · 0, 0, 3 · 0, 0, 0, -4 … Σ 1, -2, 3, -4, 5, -6, 7, -8, …
  (−∞, 0]rational roots
• A207607 → basis A007318
  1 · 3, 2 · 8, 9, 2 · 21, 33, 15, 2 … Σ 1, 5, 19, 71, 265, 989, 3691, 13775, …
  (−5, −1)nonnegative
• A008278 → (θ+1)²
  1 · 1, 4 · 1, 12, 9 · 1, 24, 63, 16 … Σ 1, 5, 22, 104, 531, 2897, 16787, …
  (−∞, 0]nonnegative
• A008277 → θ+2 (shift +1) → D
  3 · 9, 8 · 21, 48, 15 · 45, 200, 150, 24 … Σ 0, 3, 17, 84, 419, 2183, 11990, 69541, …
  (−∞, 0]nonnegative
• A367270 → x ↦ x+1
  1 · 1 · 3, 2 · 10, 12, 3 … Σ 1, 1, 5, 25, 129, 681, 3653, 19825, …
  (−∞, −1]nonnegative
• A207824: Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).
  1 · 5, 1 · 24, 10, 1 · 115, 73, 15, 1 … Σ 1, 6, 35, 204, 1189, 6930, 40391, …
  (−7, −3)nonnegative
• A001263: Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.
  1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 … Σ 1, 2, 5, 14, 42, 132, 429, 1430, …
  (−∞, 0)nonnegative
• A137320 → (θ+1)²
  1 · 0, 8 · 0, 8, 36 · 0, 16, 108, 128 … Σ 1, 8, 44, 252, 1612, 11624, 94040, …
  (−∞, 0]nonnegative
• A114656 → reverse
  1 · 1, 2 · 1, 6, 4 · 1, 12, 24, 8 … Σ 1, 3, 11, 45, 197, 903, 4279, 20793, …
  (−∞, 0]nonnegative
• A013622: Triangle of coefficients in expansion of (3+5x)^n.
  1 · 3, 5 · 9, 30, 25 · 27, 135, 225, 125 … Σ 1, 8, 64, 512, 4096, 32768, 262144, …
  (−∞, 0)[−1, 0]nonnegative+1
• A120909 → D+1
  3 · 9, 6 · 15, 36, 12 · 21, 90, 108, 24 … Σ 3, 15, 63, 243, 891, 3159, 10935, …
  (−∞, 0]nonnegativerational roots
• A207823 → D
  1 · 8, 2 · 46, 24, 3 · 232, 186, 48, 4 … Σ 0, 1, 10, 73, 470, 2828, 16310, 91371, …
  (−∞, 0]nonnegative
• A038208 → D+1
  1 · 4, 2 · 12, 16, 4 · 32, 72, 48, 8 … Σ 1, 6, 32, 160, 768, 3584, 16384, 73728, …
  (−∞, 0]nonnegativerational roots
• A125662 → reverse
  1 · 1, 3 · 1, 6, 8 · 1, 9, 25, 21 … Σ 1, 4, 15, 56, 209, 780, 2911, 10864, …
  (−1, 0]nonnegative