preserver_image / euler_polynomial, shift 0

Image under a real-rootedness preserver. All operations

352 sequences

• A028326 → θ+1
  2 · 2, 4 · 2, 8, 6 · 2, 12, 18, 8 … row sums 2, 6, 16, 40, 96, 224, 512, 1152, …
  (−∞, 0] nonnegative rational roots
• A063007 → θ+1
  1 · 1, 4 · 1, 12, 18 · 1, 24, 90, 80 … row sums 1, 5, 31, 195, 1221, 7593, 46915, …
  (−∞, 0] nonnegative
• A178820 → θ+1
  1 · 4, 8 · 10, 40, 30 · 20, 120, 180, 80 … row sums 1, 12, 80, 400, 1680, 6272, 21504, …
  (−∞, 0] nonnegative rational roots
• A059473 → θ+1
  1 · 2, 4 · 4, 20, 12 · 8, 64, 96, 32 … row sums 1, 6, 36, 200, 1068, 5544, 28192, …
  (−∞, 0] nonnegative
• A128099 → (θ+1)²
  1 · 1 · 1, 8 · 1, 16 … row sums 1, 1, 9, 17, 61, 141, 385, 921, …
  (−∞, 0] nonnegative
• A007318 → 2θ+1 → (θ+1)²
  1 · 1, 12 · 1, 24, 45 · 1, 36, 135, 112 … row sums 1, 13, 70, 284, 992, 3152, 9376, 26560, …
  (−∞, 0] nonnegative
• A172431 → ⊙ C(n,k+1) (shift -1) → θ+1
  1 · 1, 2 · 1, 8, 3 · 1, 18, 30, 4 … row sums 1, 3, 12, 53, 244, 1152, 5531, 26875, …
  (−∞, 0] nonnegative
• A123519 → θ+1
  1, 12, 12 · 1, 24, 60, 32 · 1, 40, 180, 224, 80 … row sums 25, 117, 525, 2289, 9773, 41065, …
  (−∞, 0] nonnegative
• A007318 → D → (θ+1)²
  1 · 2, 8 · 3, 24, 27 · 4, 48, 108, 64 … row sums 0, 1, 10, 54, 224, 800, 2592, 7840, …
  (−∞, 0] nonnegative
• A038220 → θ+1
  1 · 3, 4 · 9, 24, 12 · 27, 108, 108, 32 … row sums 1, 7, 45, 275, 1625, 9375, 53125, …
  (−∞, 0] nonnegative rational roots
• A013609 → (θ+1)²
  1 · 1, 8 · 1, 16, 36 · 1, 24, 108, 128 … row sums 1, 9, 53, 261, 1161, 4833, 19197, …
  (−∞, 0] nonnegative
• A060920 → θ+1
  1 · 2, 2 · 5, 10, 3 · 13, 40, 27, 4 … row sums 1, 4, 18, 84, 390, 1784, 8042, 35804, …
  (−∞, ∞) nonnegative
• A347823: Triangle read by rows: T(n,k) = (n+k+1)*binomial(n,k), 0 <= k <= n.
  A007318 → θ+1+n
  1 · 2, 3 · 3, 8, 5 · 4, 15, 18, 7 … row sums 1, 5, 16, 44, 112, 272, 640, 1472, …
  A347823 (−∞, 0] nonnegative rational roots
• A038208 → (θ+1)²
  1 · 2, 8 · 4, 32, 36 · 8, 96, 216, 128 … row sums 1, 10, 72, 448, 2560, 13824, 71680, …
  (−∞, 0] nonnegative
• A152198 → (θ+1)²
  1 · 1 · 1, 4 · 1, 4 … row sums 1, 1, 5, 5, 18, 18, 56, 56, …
  (−∞, 0] nonnegative
• A011973 → basis A007318 → (θ+1)²
  1 · 1 · 2, 4 · 3, 8 … row sums 1, 1, 6, 11, 34, 75, 190, 435, …
  (−∞, 0] nonnegative
• A081577 → θ+1
  1 · 1, 2 · 1, 8, 3 · 1, 14, 21, 4 … row sums 1, 3, 12, 40, 132, 420, 1312, 4032, …
  (−∞, 0] nonnegative
• A108838 → ⊙ C(n+1,k+1) (shift +1) → θ+1
  1 · 1, 2 · 1, 8, 3 · 1, 20, 30, 4 … row sums 1, 3, 12, 55, 276, 1477, 8296, 48393, …
  (−∞, 0] nonnegative
• A102365 → (θ+1)²
  1 · 1 · 1, 4 · 1, 20, 9 … row sums 1, 1, 5, 30, 224, 2011, 21057, 251719, …
  (−∞, 0] nonnegative
• type_b_eulerian → γ (scale 1/4) → (θ+1)²
  1 · 1 · 1, 4 · 1, 20 … row sums 1, 1, 5, 21, 118, 782, 6004, 52332, …
  (−∞, 0] nonnegative
• A173018 → reverse (shift +1) → θ+1
  1 · 0, 2 · 0, 2, 3 · 0, 2, 12, 4 … row sums 1, 2, 5, 18, 84, 480, 3240, 25200, …
  (−∞, 0] nonnegative
• basis-map coefficients of A113214, shifted → θ+1
  1 · -1, -2 · 1, 4, 3 · -1, -6, -9, -4 … row sums 1, -3, 8, -20, 48, -112, 256, -576, …
  (−∞, 0] rational roots
• A038207 → θ+1
  1 · 2, 2 · 4, 8, 3 · 8, 24, 18, 4 … row sums 1, 4, 15, 54, 189, 648, 2187, 7290, …
  (−∞, 0] nonnegative rational roots
• A094816 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 16, 18, 4 … row sums 1, 3, 10, 39, 181, 995, 6386, 47075, …
  (−∞, 0] nonnegative
• A008277 → (θ+1)²
  1 · 1, 4 · 1, 12, 9 · 1, 28, 54, 16 … row sums 1, 5, 22, 99, 471, 2386, 12867, 73681, …
  (−∞, 0] nonnegative
• A013611 → θ+1
  1 · 1, 8 · 1, 16, 48 · 1, 24, 144, 256 … row sums 1, 9, 65, 425, 2625, 15625, 90625, …
  (−∞, 0] nonnegative rational roots
• type_b_eulerian → γ (scale 1/4) → θ+1
  1 · 1 · 1, 2 · 1, 10 … row sums 1, 1, 3, 11, 52, 300, 2040, 15960, …
  (−∞, 0] nonnegative
• A232535 → θ+1
  1 · 1, 4 · 1, 16, 9 · 1, 36, 75, 16 … row sums 1, 5, 26, 128, 608, 2816, 12800, 57344, …
  (−∞, 0] nonnegative
• A038220 → (θ+1)²
  1 · 3, 8 · 9, 48, 36 · 27, 216, 324, 128 … row sums 1, 11, 93, 695, 4825, 31875, 203125, …
  (−∞, 0] nonnegative
• basis-map coefficients of A021012 → θ+1
  1 · 0, -2 · 0, 0, 6 · 0, 0, 0, -24 … row sums 1, -2, 6, -24, 120, -720, 5040, -40320, …
  (−∞, 0] rational roots
• A008292 → γ (shift +1) → (θ+1)²
  1 · 1 · 1, 8 · 1, 32 … row sums 1, 1, 9, 33, 233, 1433, 11289, 92097, …
  (−∞, 0] nonnegative
• A008292 → γ (shift +1) → θ+1
  1 · 1 · 1, 4 · 1, 16 … row sums 1, 1, 5, 17, 93, 513, 3477, 25569, …
  (−∞, 0] nonnegative
• A173018 → (θ+1)²
  1 · 1 · 1, 4 · 1, 16, 9 … row sums 1, 1, 5, 26, 160, 1140, 9240, 84000, …
  (−∞, 0] nonnegative
• A132393 → (θ+1)²
  1 · 0, 4 · 0, 4, 9 · 0, 8, 27, 16 … row sums 1, 4, 13, 51, 244, 1392, 9260, 70508, …
  (−∞, 0] nonnegative
• A011973 → (θ+1)²
  1 · 1 · 1, 4 · 1, 8 … row sums 1, 1, 5, 9, 22, 44, 91, 179, …
  (−∞, 0] nonnegative
• A062264 → θ+1
  1 · 1, 10 · 1, 24, 45 · 1, 42, 189, 140 … row sums 1, 11, 70, 372, 1815, 8437, 38038, …
  (−∞, 0] nonnegative
• A159854 → (θ+1)²
  0, 0, 9 · 0, 0, 9, 16 · 0, 0, 9, 32, 25 … row sums 9, 25, 66, 168, 416, 1008, 2400, 5632, …
  (−∞, 0] nonnegative
• A055356 → (θ+1)²
  1 · 1 · 1, 4 · 1, 16, 18 … row sums 1, 1, 5, 35, 303, 3123, 37349, 508547, …
  (−∞, 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
• A081581 → θ+1
  1 · 1, 2 · 1, 16, 3 · 1, 30, 45, 4 … row sums 1, 3, 20, 80, 372, 1540, 6496, 26496, …
  (−∞, 0] nonnegative
• A086645 → (θ+1)²
  1 · 1, 4 · 1, 24, 9 · 1, 60, 135, 16 … row sums 1, 5, 34, 212, 1216, 6592, 34304, …
  (−∞, 0] nonnegative
• A060920 → (θ+1)²
  1 · 2, 4 · 5, 20, 9 · 13, 80, 81, 16 … row sums 1, 6, 34, 190, 1026, 5342, 26962, …
  (−∞, ∞) nonnegative
• A039755 → θ+1
  1 · 1, 2 · 1, 8, 3 · 1, 26, 27, 4 … row sums 1, 3, 12, 58, 324, 2044, 14320, 109960, …
  (−∞, 0] nonnegative
• A059344 → ⊙ 1/k! (shift -1) → (θ+1)²
  1 · 1 · 1, 8 · 1, 24 … row sums 1, 1, 9, 25, 103, 351, 1251, 4299, …
  (−∞, 0] nonnegative
• A143685 → θ+1
  1 · 1, 2 · 1, 22, 3 · 1, 42, 63, 4 … row sums 1, 3, 26, 110, 615, 2821, 13828, 63756, …
  (−∞, 0] nonnegative
• A013611 → (θ+1)²
  1 · 1, 16 · 1, 32, 144 · 1, 48, 432, 1024 … row sums 1, 17, 177, 1505, 11425, 80625, 540625, …
  (−∞, 0] nonnegative
• A145597 → θ+1
  1 · 3, 6 · 6, 30, 18 · 10, 90, 135, 40 … row sums 1, 9, 54, 275, 1287, 5733, 24752, …
  (−∞, 0] nonnegative
• A001263 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 12, 18, 4 … row sums 1, 3, 10, 35, 126, 462, 1716, 6435, …
  (−∞, 0] nonnegative
• A207823 → (θ+1)²
  1 · 4, 4 · 15, 32, 9 · 56, 184, 108, 16 … row sums 1, 8, 56, 364, 2255, 13504, 78841, …
  (−∞, 0] nonnegative
• A007318 → 2θ+1 → θ+1
  1 · 1, 6 · 1, 12, 15 · 1, 18, 45, 28 … row sums 1, 7, 28, 92, 272, 752, 1984, 5056, …
  (−∞, 0] nonnegative