preserver_image / euler_polynomial, shift 0

Image under a real-rootedness preserver. All operations

352 sequences

• A207606 → θ+1
  4, 14, 6 · 5, 32, 33, 8 · 6, 60, 108, 60, 10 … row sums 24, 78, 244, 743, 2218, 6521, 18942, …
  (−∞, 0] nonnegative
• A102662 → (θ+1)²
  1 · 1, 12 · 1, 20, 27 · 1, 28, 99, 48 … row sums 1, 13, 48, 176, 591, 1895, 5858, 17614, …
  (−∞, 0] nonnegative
• A085880 → θ+1
  1 · 1, 2 · 2, 8, 6 · 5, 30, 45, 20 … row sums 1, 3, 16, 100, 672, 4704, 33792, …
  (−∞, 0] nonnegative rational roots
• A081577 → (θ+1)²
  1 · 1, 4 · 1, 16, 9 · 1, 28, 63, 16 … row sums 1, 5, 26, 108, 424, 1564, 5552, 19104, …
  (−∞, 0] nonnegative
• A011973 → x ↦ x−1/4 → θ+1
  1 · 2 · 3, 8 · 4, 32 … row sums 1, 2, 11, 36, 133, 454, 1551, 5192, …
  (−∞, 0] nonnegative
• A102547 → reverse → θ+1
  0, 2 · 0, 0, 3 · 0, 0, 0, 4 · 0, 0, 0, 4, 5 … row sums 2, 3, 4, 9, 16, 25, 42, 70, …
  (−∞, 0] nonnegative
• A113214 → (θ+1)²
  1 · 2, 4 · 0, 12, 9 · 0, 8, 36, 16 … row sums 1, 6, 21, 60, 150, 348, 763, 1608, …
  (−∞, 0] nonnegative
• A078812 → θ+1
  1 · 2, 2 · 3, 8, 3 · 4, 20, 18, 4 … row sums 1, 4, 14, 46, 145, 444, 1331, 3926, …
  (−∞, 0] nonnegative
• A227450 → (θ+1)²
  2 · 2, 16 · 2, 48, 72 · 2, 112, 432, 256 … row sums 2, 18, 122, 802, 5402, 37922, 278938, …
  (−∞, 0] nonnegative
• A007318 → (1/2+(1/2)n)θ+1+n → (θ+1)²
  1 · 2, 12 · 3, 36, 54 · 4, 72, 216, 160 … row sums 1, 14, 93, 452, 1840, 6672, 22288, …
  (−∞, 0] nonnegative
• A304236 → (θ+1)²
  1 · 1 · 1, 12 · 1, 24 … row sums 1, 1, 13, 25, 118, 292, 979, 2611, …
  (−∞, 0] nonnegative
• A008292 → γ (scale 1/2) (shift +1) → (θ+1)²
  1 · 1 · 1, 4 · 1, 16 … row sums 1, 1, 5, 17, 81, 411, 2393, 15329, …
  (−∞, 0] nonnegative
• A038207 → (θ+1)²
  1 · 2, 4 · 4, 16, 9 · 8, 48, 54, 16 … row sums 1, 6, 29, 126, 513, 1998, 7533, 27702, …
  (−∞, 0] nonnegative
• A114687 → (θ+1)²
  1 · 1, 8 · 1, 24, 36 · 1, 48, 216, 128 … row sums 1, 9, 61, 393, 2481, 15473, 95677, …
  (−∞, 0] nonnegative
• A145597 → (θ+1)²
  1 · 3, 12 · 6, 60, 54 · 10, 180, 405, 160 … row sums 1, 15, 120, 755, 4191, 21567, 105560, …
  (−∞, 0] nonnegative
• A172431 → (θ+1)²
  1 · 1, 8 · 1, 16, 27 · 1, 24, 90, 64 … row sums 1, 9, 44, 179, 667, 2352, 7973, 26229, …
  (−∞, 0] nonnegative
• A062190 → θ+1
  1 · 1, 12 · 1, 28, 63 · 1, 48, 252, 224 … row sums 1, 13, 92, 525, 2695, 13013, 60424, …
  (−∞, 0] nonnegative
• A173018 → θ+1
  1 · 1 · 1, 2 · 1, 8, 3 … row sums 1, 1, 3, 12, 60, 360, 2520, 20160, …
  (−∞, 0] nonnegative
• A154986 → θ+1
  1 · 1, 2 · 1, 8, 3 · 1, 22, 33, 4 … row sums 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  (−∞, 0] nonnegative
• A304236 → θ+1
  1 · 1 · 1, 6 · 1, 12 … row sums 1, 1, 7, 13, 46, 106, 301, 739, …
  (−∞, 0] nonnegative
• A227450 → θ+1
  2 · 2, 8 · 2, 24, 24 · 2, 56, 144, 64 … row sums 2, 10, 50, 266, 1522, 9354, 61490, …
  (−∞, 0] nonnegative
• A143495 → (θ+1)²
  1 · 3, 4 · 9, 28, 9 · 27, 148, 108, 16 … row sums 1, 7, 46, 299, 1967, 13248, 91903, …
  (−∞, 0] nonnegative
• A103633 → (θ+1)²
  1 · 0, 4 · 0, 4, 9 · 0, 0, 9, 16 … row sums 1, 4, 13, 25, 66, 102, 248, 344, …
  (−∞, 0] nonnegative
• A039755 → (θ+1)²
  1 · 1, 4 · 1, 16, 9 · 1, 52, 81, 16 … row sums 1, 5, 26, 150, 964, 6836, 52936, …
  (−∞, 0] nonnegative
• A135838 → (θ+1)²
  1 · 2, 8 · 2, 16, 18 · 4, 48, 108, 64 … row sums 1, 10, 36, 224, 640, 3456, 8960, 45056, …
  (−∞, 0] nonnegative
• basis-map coefficients of A021012 → (θ+1)²
  1 · 0, -4 · 0, 0, 18 · 0, 0, 0, -96 … row sums 1, -4, 18, -96, 600, -4320, 35280, …
  (−∞, 0] rational roots
• basis-map coefficients of A026729 → basis A056242 → θ+1
  1 · 1, 4 · 1, 10, 12 · 1, 18, 48, 32 … row sums 1, 5, 23, 99, 408, 1632, 6388, 24596, …
  (−∞, 0] nonnegative
• A023531 → (θ+1)² → (θ+1)²
  1 · 0, 16 · 0, 0, 81 · 0, 0, 0, 256 … row sums 1, 16, 81, 256, 625, 1296, 2401, 4096, …
  (−∞, 0] nonnegative rational roots
• A008288 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 10, 15, 4 … row sums 1, 3, 10, 30, 87, 245, 676, 1836, …
  (−∞, 0] nonnegative
• A062196 → (θ+1)²
  1 · 1, 12 · 1, 32, 54 · 1, 60, 270, 160 … row sums 1, 13, 87, 491, 2562, 12762, 61611, …
  (−∞, 0] nonnegative
• A143494 → (θ+1)²
  1 · 2, 4 · 4, 20, 9 · 8, 76, 81, 16 … row sums 1, 6, 33, 181, 1020, 5977, 36589, …
  (−∞, 0] nonnegative
• A062145 → θ+1
  1 · 1, 8 · 1, 20, 30 · 1, 36, 135, 80 … row sums 1, 9, 51, 252, 1170, 5247, 23023, …
  (−∞, 0] nonnegative
• A347056: Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.
  A007318 → (1/3+(1/2)n+(1/6)n²)θ+1+(3/2)n+(1/2)n²
  1 · 3, 4 · 6, 16, 10 · 10, 40, 50, 20 … row sums 1, 7, 32, 120, 400, 1232, 3584, 9984, …
  A347056 (−∞, 0] nonnegative rational roots
• A274498 → (θ+1)²
  1 · 3 · 3, 24 · 9, 72 … row sums 1, 3, 27, 81, 477, 1431, 7047, 21141, …
  (−∞, 0] nonnegative
• A143683 → θ+1
  1 · 1, 2 · 1, 20, 3 · 1, 38, 57, 4 … row sums 1, 3, 24, 100, 528, 2352, 11008, 48960, …
  (−∞, 0] nonnegative
• A120909 → (θ+1)²
  3 · 3, 24 · 3, 48, 108 · 3, 72, 324, 384 … row sums 3, 27, 159, 783, 3483, 14499, 57591, …
  (−∞, 0] nonnegative
• A123519 → (θ+1)²
  1, 24, 36 · 1, 48, 180, 128 · 1, 80, 540, 896, 400 … row sums 61, 357, 1917, 9717, 47249, 222633, …
  (−∞, 0] nonnegative
• delannoy → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 10, 15, 4 … row sums 1, 3, 10, 30, 87, 245, 676, 1836, …
  (−∞, 0] nonnegative
• A063967 → θ+1
  1 · 1, 2 · 2, 6, 3 · 3, 14, 15, 4 … row sums 1, 3, 11, 36, 116, 364, 1124, 3424, …
  (−∞, 0] nonnegative
• A334218 → θ+1
  1 · 1 · 0, 4 · 0, 6, 9 … row sums 1, 1, 4, 15, 72, 420, 2880, 22680, …
  (−∞, 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
• A008292 → γ (scale 1/4) (shift +1) → θ+1
  1 · 2 · 4, 4 · 8, 32 … row sums 1, 2, 8, 40, 240, 1680, 13440, 120960, …
  (−∞, 0] nonnegative
• A344565 → (θ+1)²
  3 · 15, 60 · 45, 360, 405 · 105, 1260, 2835, 1680 … row sums 3, 75, 810, 5880, 33600, 163296, …
  (−∞, 0] nonnegative
• basis-map coefficients of A026729, shifted → (θ+1)²
  1 · 0, -4 · 0, 0, 9 · 0, 0, 0, -16 … row sums 1, -4, 9, -16, 25, -36, 49, -64, …
  (−∞, 0] rational roots
• A274496 → θ+1
  1 · 2 · 2, 4 · 4, 8 … row sums 1, 2, 6, 12, 32, 64, 160, 320, …
  (−∞, 0] 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
• A081578 → θ+1
  1 · 1, 2 · 1, 10, 3 · 1, 18, 27, 4 … row sums 1, 3, 14, 50, 183, 637, 2188, 7380, …
  (−∞, 0] nonnegative
• A082985 → θ+1
  1 · 1, 6 · 1, 10, 15 · 1, 14, 42, 28 … row sums 1, 7, 26, 85, 265, 804, 2393, 7019, …
  (−∞, 0] nonnegative
• A030528 → (θ+1)²
  1 · 1, 4 · 0, 8, 9 · 0, 4, 27, 16 … row sums 1, 5, 17, 47, 116, 266, 579, 1213, …
  (−∞, 0] nonnegative
• A159854 → θ+1
  0, 0, 3 · 0, 0, 3, 4 · 0, 0, 3, 8, 5 … row sums 3, 7, 16, 36, 80, 176, 384, 832, …
  (−∞, 0] nonnegative rational roots