Proof methods / preserver_image

Image under a real-rootedness preserver

1,387 sequences

• A088996: Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.
  1 · 0, 1 · 0, 1, 2 · 0, 2, 7, 6 … Σ 1, 1, 3, 15, 105, 945, 10395, 135135, …
  [−1, 0]nonnegativerational roots
• A008292 → D
  1 · 4, 2 · 11, 22, 3 · 26, 132, 78, 4 … Σ 0, 1, 6, 36, 240, 1800, 15120, 141120, …
  (−∞, 0]nonnegative
• A099097 → D+1
  1 · 3, 3 · 1, 19, 9 · 0, 12, 87, 27 … Σ 1, 6, 29, 126, 516, 2034, 7807, 29382, …
  (−∞, 0]nonnegative
• A102547 → reverse → x ↦ x/(x+1)
  0, 1 · 0, 0, 1 · 0, 0, 0, 1 · 0, 0, 0, 1, 2 … Σ 1, 1, 1, 3, 5, 7, 13, 23, …
  [−1, 0]nonnegative
• A185411: A triangular decomposition of the double factorial numbers A001147.
  A185410 → reverse
  1 · 0, 1 · 0, 2, 1 · 0, 4, 10, 1 … Σ 1, 1, 3, 15, 105, 945, 10395, 135135, …
  (−∞, 0]nonnegative
• A013610 → D+1
  1 · 4, 3 · 7, 24, 9 · 10, 63, 108, 27 … Σ 1, 7, 40, 208, 1024, 4864, 22528, …
  (−∞, 0]nonnegativerational roots
• A132812 → D
  2 · 9, 6 · 24, 48, 12 · 50, 200, 150, 20 … Σ 0, 2, 15, 84, 420, 1980, 9009, 40040, …
  (−∞, 0]nonnegative
• A008288 → basis A007318
  1 · 2, 1 · 5, 5, 1 · 12, 18, 8, 1 … Σ 1, 3, 11, 39, 139, 495, 1763, 6279, …
  (−∞, −1]nonnegative
• A030528 → D
  1 · 2, 2 · 1, 6, 3 · 0, 6, 12, 4 … Σ 0, 1, 4, 10, 22, 45, 88, 167, …
  (−∞, 0]nonnegative
• A347056 → x ↦ x/(x+1)
  1 · 3, 7 · 6, 28, 32 · 10, 70, 160, 120 … Σ 1, 10, 66, 360, 1755, 7938, 34020, …
  [−1, 0]nonnegativerational roots
• A102365 → x ↦ x+1
  1 · 1 · 2, 1 · 7, 7, 1 … Σ 1, 1, 3, 15, 105, 945, 10395, 135135, …
  (−∞, −1]nonnegative
• A113953 → x ↦ x−1
  1 · -1, 1 · -1, 0, 1 · 3, -5, 1, 1 … Σ 1, 0, 0, 0, 0, 0, 0, 0, …
  (−7, 1]
• A108299 → D
  -1 · -1, -2 · -1, -4, 3 · -1, -6, 6, 4 … Σ 0, -1, -3, -2, 3, 7, 4, -5, …
  (−∞, ∞)
• A154986 → D+1
  1 · 2, 1 · 5, 6, 1 · 12, 33, 14, 1 … Σ 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  (−∞, 0]nonnegative
• A038207 → D+1
  1 · 3, 1 · 8, 6, 1 · 20, 24, 9, 1 … Σ 1, 4, 15, 54, 189, 648, 2187, 7290, …
  (−∞, 0]nonnegativerational roots
• A102365 → D
  1 · 5, 2 · 18, 30, 3 · 58, 258, 111, 4 … Σ 0, 0, 1, 7, 51, 431, 4208, 46824, …
  (−∞, 0]nonnegative
• reversal of Riordan array (1+2x,x(1+x)).
  1 · 1, 2 · 1, 3 · 1, 4, 2 … Σ 1, 3, 4, 7, 11, 18, 29, 47, …
  (−∞, 0)nonnegative
• A113214 → x ↦ x−3
  1 · -1, 1 · 0, -3, 1 · 3, 5, -5, 1 … Σ 1, 0, -2, 4, -4, 0, 8, -16, …
  (−1, 3]
• A103327 → basis A007318
  1 · 4, 1 · 16, 12, 1 · 64, 80, 24, 1 … Σ 1, 5, 29, 169, 985, 5741, 33461, …
  (−∞, −1]nonnegative
• A059344 → basis A007318
  1 · 1 · 3, 2 · 7, 6 … Σ 1, 1, 5, 13, 73, 281, 1741, 8485, …
  (−∞, −1]nonnegative
• A049458: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 2.
  1 · -3, 1 · 12, -7, 1 · -60, 47, -12, 1 … Σ 1, -2, 6, -24, 120, -720, 5040, -40320, …
  [5/2, ∞)rational roots
• A108838 → D+1
  2 · 5, 2 · 12, 12, 2 · 25, 50, 21, 2 … Σ 2, 7, 26, 98, 372, 1419, 5434, 20878, …
  (−∞, 0]nonnegative
• A106246 → reverse
  1 · 1, 2 · 1, 4, 1 · 1, 6, 3 … Σ 1, 3, 6, 10, 15, 21, 28, 36, …
  (−∞, 0]nonnegative
• A159854 → x ↦ x/(x+1)
  0, 0, 1 · 0, 0, 1, 2 · 0, 0, 1, 4, 4 … Σ 1, 3, 9, 27, 81, 243, 729, 2187, …
  [−1, 0]nonnegativerational roots
• A099093 → x ↦ x−15
  1 · -45, 3 · 1980, -267, 9 · -87075, 17685, -1197, 27 … Σ 1, -42, 1722, -70560, 2891196, …
  (41/3, 15]
• A094638 → D
  1 · 3, 4 · 6, 22, 18 · 10, 70, 150, 96 … Σ 0, 1, 7, 46, 326, 2556, 22212, 212976, …
  (−∞, 0]nonnegative
• A178820 → (θ+1)²
  1 · 4, 16 · 10, 80, 90 · 20, 240, 540, 320 … Σ 1, 20, 180, 1120, 5600, 24192, 94080, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729 → (θ+1)²
  1 · 0, 4 · 0, 0, 9 · 0, 0, 0, 16 … Σ 1, 4, 9, 16, 25, 36, 49, 64, …
  (−∞, 0]nonnegativerational roots
• A102662 → θ+1
  1 · 1, 6 · 1, 10, 9 · 1, 14, 33, 12 … Σ 1, 7, 20, 60, 171, 477, 1306, 3526, …
  (−∞, 0]nonnegative
• A207823 → reverse
  1 · 1, 4 · 1, 8, 15 · 1, 12, 46, 56 … Σ 1, 5, 24, 115, 551, 2640, 12649, 60605, …
  (−1/2, 0]nonnegative
• A094816 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 16, 23, 9, 1 … Σ 1, 3, 11, 49, 261, 1631, 11743, 95901, …
  (−∞, −1]nonnegative
• A049459: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 3.
  1 · -4, 1 · 20, -9, 1 · -120, 74, -15, 1 … Σ 1, -3, 12, -60, 360, -2520, 20160, …
  [7/2, ∞)rational roots
• A331333: Interpolating the factorial and the powers of 2. Triangle read by rows, T(n, k) for 0 <= k <= n.
  1 · 1, 2 · 2, 8, 4 · 6, 36, 36, 8 … Σ 1, 3, 14, 86, 648, 5752, 58576, 671568, …
  (−∞, 0)nonnegative
• A116414 → (θ+1)²
  1 · 4, 4 · 13, 32, 9 · 40, 168, 108, 16 … Σ 1, 8, 54, 332, 1921, 10656, 57301, …
  (−∞, 0]nonnegative
• A143495 → D
  1 · 7, 2 · 37, 24, 3 · 175, 194, 54, 4 … Σ 0, 1, 9, 64, 427, 2821, 18890, 129683, …
  (−∞, 0]nonnegative
• A008277 → θ+1 → D+1
  1 · 3, 2 · 7, 12, 3 · 15, 50, 30, 4 … Σ 1, 5, 22, 99, 471, 2386, 12867, 73681, …
  (−∞, 0]nonnegative
• A113214 → x ↦ x−15
  1 · -13, 1 · 180, -27, 1 · -2505, 557, -41, 1 … Σ 1, -12, 154, -1988, 25676, -331632, …
  (11, 15]
• A391403: Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).
  1 · 1 · 1, 2 · 1, 6 … Σ 1, 1, 3, 7, 19, 51, 141, 393, …
  (−∞, 0)nonnegative
• A034870 → x ↦ x+1
  1 · 4, 4, 1 · 16, 32, 24, 8, 1 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, −1]nonnegativerational roots
• A125092 → x ↦ x/(x+1)
  1 · 1, 5 · 1, 10, 18 · 1, 15, 54, 56 … Σ 1, 6, 29, 126, 513, 1998, 7533, 27702, …
  [−1, 0]nonnegative
• A135838 → D+1
  1 · 4, 2 · 6, 8, 2 · 16, 36, 24, 4 … Σ 1, 6, 16, 80, 192, 896, 2048, 9216, …
  (−∞, 0]nonnegativerational roots
• A039755 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 16, 38, 24 … Σ 1, 3, 13, 79, 601, 5339, 53861, 607527, …
  [−1, 0]nonnegative
• A152198 → x ↦ x+1
  1 · 1 · 2, 1 · 2, 1 … Σ 1, 1, 3, 3, 9, 9, 27, 27, …
  (−∞, −1]nonnegativerational roots
• A120910 → D+1
  3 · 9, 3 · 24, 18, 3 · 60, 72, 27, 3 … Σ 3, 12, 45, 162, 567, 1944, 6561, 21870, …
  (−∞, 0]nonnegativerational roots
• A122431 → θ+1
  1, 6, 9, 4 · 0, 2, 9, 12, 5 · 0, 0, 3, 12, 15, 6 … Σ 20, 28, 36, 44, 52, 60, 68, 76, …
  (−∞, 0]nonnegativerational roots
• A055356 → x ↦ x+1
  1 · 1 · 2, 1 · 7, 8, 2 … Σ 1, 1, 3, 17, 143, 1597, 22243, 371241, …
  (−∞, −1]nonnegative
• A103633 → D
  1 · 1, 2 · 0, 2, 3 · 0, 2, 6, 4 … Σ 0, 1, 3, 5, 12, 16, 36, 44, …
  (−∞, 0]nonnegativerational roots
• A085880 → x ↦ x+1
  1 · 2, 1 · 8, 8, 2 · 40, 60, 30, 5 … Σ 1, 3, 18, 135, 1134, 10206, 96228, …
  (−∞, −1]nonnegativerational roots
• A063967 → (θ+1)²
  1 · 1, 4 · 2, 12, 9 · 3, 28, 45, 16 … Σ 1, 5, 23, 92, 346, 1240, 4296, 14496, …
  (−∞, 0]nonnegative
• A013611 → D
  4 · 8, 32 · 12, 96, 192 · 16, 192, 768, 1024 … Σ 0, 4, 40, 300, 2000, 12500, 75000, …
  (−∞, 0]nonnegativerational roots