preserver_image / derivative_polynomial, shift 0

Image under a real-rootedness preserver. All operations

410 sequences

• A126216 → D+1
  1 · 3, 1 · 10, 7, 1 · 35, 39, 12, 1 … row sums 1, 4, 18, 87, 439, 2278, 12052, 64669, …
  (−∞, 0] nonnegative
• A106800 → D
  1 · 3, 2 · 6, 14, 3 · 10, 50, 45, 4 … row sums 0, 0, 1, 5, 23, 109, 544, 2876, …
  (−∞, 0] nonnegative
• A010054 → reverse → D+1
  1 · 4, 4 · 0, 14, 7 · 0, 0, 30, 10 … row sums 1, 8, 21, 40, 65, 96, 133, 176, …
  (−∞, 0] nonnegative rational roots
• A137597: Triangle read by rows: A008277 * A007318.
  A007318 → basis A049020
  1 · 2, 1 · 5, 5, 1 · 15, 22, 9, 1 … row sums 1, 3, 11, 47, 227, 1215, 7107, 44959, …
  A137597 (−∞, −1] nonnegative
• A267482 → D
  1 · 1, 2 · -2, 2, 3 · -2, -6, 3, 4 … row sums 0, 1, 3, 3, -1, -6, -6, 1, …
  (−∞, ∞)
• A013611 → D+1
  1 · 5, 4 · 9, 40, 16 · 13, 108, 240, 64 … row sums 1, 9, 65, 425, 2625, 15625, 90625, …
  (−∞, 0] nonnegative rational roots
• A081580 → D+1
  1 · 2, 1 · 8, 9, 1 · 14, 39, 16, 1 … row sums 1, 3, 18, 70, 303, 1197, 4756, 18396, …
  (−∞, 0] nonnegative
• A267482 → D+1
  1 · 2, 1 · 0, 3, 1 · -3, 0, 4, 1 … row sums 1, 3, 4, 2, -3, -7, -5, 3, …
  (−∞, ∞)
• A008277 → θ+2 (shift +1) → D
  3 · 9, 8 · 21, 48, 15 · 45, 200, 150, 24 … row sums 0, 3, 17, 84, 419, 2183, 11990, 69541, …
  (−∞, 0] nonnegative
• A120909 → D+1
  3 · 9, 6 · 15, 36, 12 · 21, 90, 108, 24 … row sums 3, 15, 63, 243, 891, 3159, 10935, …
  (−∞, 0] nonnegative rational roots
• A207823 → D
  1 · 8, 2 · 46, 24, 3 · 232, 186, 48, 4 … row sums 0, 1, 10, 73, 470, 2828, 16310, 91371, …
  (−∞, 0] nonnegative
• A038208 → D+1
  1 · 4, 2 · 12, 16, 4 · 32, 72, 48, 8 … row sums 1, 6, 32, 160, 768, 3584, 16384, 73728, …
  (−∞, 0] nonnegative rational roots
• A062264 → D
  5 · 12, 30 · 21, 126, 105 · 32, 336, 672, 280 … row sums 0, 5, 42, 252, 1320, 6435, 30030, …
  (−∞, 0] nonnegative
• A114687 → D+1
  1 · 3, 2 · 7, 14, 4 · 13, 60, 48, 8 … row sums 1, 5, 25, 129, 681, 3653, 19825, …
  (−∞, 0] nonnegative
• A060920 → D
  1 · 5, 2 · 20, 18, 3 · 71, 102, 42, 4 … row sums 0, 1, 7, 41, 219, 1101, 5311, 24881, …
  (−∞, ∞) nonnegative
• A169803 → D+1
  1 · 2, 1 · 3, 2 · 4, 5, 1 … row sums 1, 3, 5, 10, 18, 33, 59, 105, …
  (−∞, 0] nonnegative
• A116412 → D+1
  1 · 4, 1 · 12, 8, 1 · 33, 39, 12, 1 … row sums 1, 5, 21, 85, 332, 1266, 4741, 17507, …
  (−∞, 0] nonnegative
• A013609 → D+1
  1 · 3, 2 · 5, 12, 4 · 7, 30, 36, 8 … row sums 1, 5, 21, 81, 297, 1053, 3645, 12393, …
  (−∞, 0] nonnegative rational roots
• A062190 → D+1
  1 · 7, 6 · 15, 56, 21 · 25, 192, 252, 56 … row sums 1, 13, 92, 525, 2695, 13013, 60424, …
  (−∞, 0] nonnegative
• A023531 → 2D+1 → D+1
  1 · 3, 1 · 4, 6, 1 · 0, 12, 9, 1 … row sums 1, 4, 11, 22, 37, 56, 79, 106, …
  (−∞, 0] nonnegative
• A023531 → 2D+1 → D
  1 · 4, 2 · 0, 12, 3 · 0, 0, 24, 4 … row sums 0, 1, 6, 15, 28, 45, 66, 91, …
  (−∞, 0] nonnegative rational roots
• A108838 → ⊙ C(n+1,k+1) (shift +1) → D
  1 · 4, 2 · 10, 20, 3 · 20, 100, 60, 4 … row sums 0, 1, 6, 33, 184, 1055, 6222, 37639, …
  (−∞, 0] nonnegative
• A026729 → D
  1 · 1, 2 · 0, 4, 3 · 0, 2, 9, 4 … row sums 0, 1, 3, 7, 15, 30, 58, 109, …
  (−∞, 0] nonnegative
• A113214 → D+1
  1 · 3, 1 · 3, 5, 1 · 2, 10, 7, 1 … row sums 1, 4, 9, 20, 40, 78, 147, 272, …
  (−∞, 0] nonnegative
• A130123 → D+1
  1 · 2, 2 · 0, 8, 4 · 0, 0, 24, 8 … row sums 1, 4, 12, 32, 80, 192, 448, 1024, …
  (−∞, 0] nonnegative rational roots
• A038220 → D+1
  1 · 5, 2 · 21, 20, 4 · 81, 126, 60, 8 … row sums 1, 7, 45, 275, 1625, 9375, 53125, …
  (−∞, 0] nonnegative rational roots
• A113413 → D+1
  1 · 3, 1 · 6, 6, 1 · 10, 20, 9, 1 … row sums 1, 4, 13, 40, 117, 332, 921, 2512, …
  (−∞, 0] nonnegative
• A123519 → D+1
  7, 14, 4 · 13, 52, 44, 8 · 21, 140, 228, 120, 16 … row sums 25, 117, 525, 2289, 9773, 41065, …
  (−∞, 0] nonnegative
• A129533: Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.
  A007318 → (−1/2)D²+(−1/2+(1/2)n)D
  0, 1 · 0, 3, 3 · 0, 6, 12, 6 · 0, 10, 30, 30, 10 … row sums 0, 0, 1, 6, 24, 80, 240, 672, …
  A129533 (−∞, ∞) nonnegative rational roots
• A062196 → D+1
  1 · 4, 3 · 9, 20, 6 · 16, 75, 60, 10 … row sums 1, 7, 35, 161, 714, 3102, 13299, 56485, …
  (−∞, 0] nonnegative
• A178820 → D
  4 · 20, 20 · 60, 120, 60 · 140, 420, 420, 140 … row sums 0, 4, 40, 240, 1120, 4480, 16128, …
  (−∞, 0] nonnegative rational roots
• A023531 → D → D+1
  1 · 2, 2 · 0, 6, 3 · 0, 0, 12, 4 … row sums 0, 1, 4, 9, 16, 25, 36, 49, …
  (−∞, 0] nonnegative rational roots
• A113953 → D+1
  1 · 1, 1 · 2, 4, 1 · 0, 8, 7, 1 … row sums 1, 2, 7, 16, 41, 94, 219, 492, …
  (−∞, 0] nonnegative
• A249252 → D+1
  1 · 5, 4 · 9, 24, 8 · 13, 76, 80, 16 … row sums 1, 9, 41, 185, 793, 3305, 13481, 54105, …
  (−∞, 0] nonnegative
• A011973 → D
  1 · 2 · 3, 2 · 4, 6 … row sums 0, 0, 1, 2, 5, 10, 20, 38, …
  (−∞, 0] nonnegative
• A344565 → D
  15 · 90, 90 · 315, 630, 315 · 840, 2520, 2520, 840 … row sums 0, 15, 180, 1260, 6720, 30240, 120960, …
  (−∞, 0] nonnegative rational roots
• A027465 → D+1
  1 · 4, 1 · 15, 8, 1 · 54, 45, 12, 1 … row sums 1, 5, 24, 112, 512, 2304, 10240, 45056, …
  (−∞, 0] nonnegative rational roots
• A137320 → D+1
  1 · 2, 2 · 2, 10, 4 · 4, 28, 36, 8 … row sums 1, 4, 16, 76, 428, 2808, 21096, 178848, …
  (−∞, 0] nonnegative
• narayana → γ (shift +1) → D
  1 · 3 · 6, 4 · 10, 20 … row sums 0, 0, 1, 3, 10, 30, 90, 266, …
  (−∞, 0] nonnegative
• A249248 → D+1
  1 · 4, 3 · 8, 13, 3 · 13, 56, 31, 3 … row sums 1, 7, 24, 103, 419, 1815, 7970, 36277, …
  (−∞, 0] nonnegative
• A094816 → D+1
  1 · 2, 1 · 4, 5, 1 · 9, 20, 9, 1 … row sums 1, 3, 10, 39, 181, 995, 6386, 47075, …
  (−∞, 0] nonnegative
• A008277 → D
  1 · 3, 2 · 7, 12, 3 · 15, 50, 30, 4 … row sums 0, 1, 5, 22, 99, 471, 2386, 12867, …
  (−∞, 0] nonnegative
• A062190 → D
  6 · 14, 42 · 24, 168, 168 · 36, 432, 1008, 504 … row sums 0, 6, 56, 360, 1980, 10010, 48048, …
  (−∞, 0] nonnegative
• A008292 → γ (scale 1/4) (shift +1) → D
  2 · 16 · 88, 32 · 416, 544 … row sums 0, 0, 2, 16, 120, 960, 8400, 80640, …
  (−∞, 0] nonnegative
• A085478 → D
  1 · 3, 2 · 6, 10, 3 · 10, 30, 21, 4 … row sums 0, 0, 1, 5, 19, 65, 210, 654, …
  (−∞, 0] nonnegative
• A100861 → D+1
  1 · 1 · 2, 1 · 4, 3 … row sums 1, 1, 3, 7, 22, 66, 226, 778, …
  (−∞, 0] nonnegative
• basis-map coefficients of A030528, shifted → D
  -1 · 0, 2 · 0, 0, -3 · 0, 0, 0, 4 … row sums 0, -1, 2, -3, 4, -5, 6, -7, …
  (−∞, 0] rational roots
• basis-map coefficients of A030528, shifted → D+1
  1 · -1, -1 · 0, 2, 1 · 0, 0, -3, -1 … row sums 1, -2, 3, -4, 5, -6, 7, -8, …
  (−∞, 0] rational roots
• A114656 → D
  1 · 6, 2 · 24, 24, 3 · 80, 160, 60, 4 … row sums 0, 1, 8, 51, 304, 1765, 10128, 57799, …
  (−∞, 0] nonnegative
• A102662 → D+1
  1 · 4, 3 · 6, 11, 3 · 8, 29, 20, 3 … row sums 1, 7, 20, 60, 171, 477, 1306, 3526, …
  (−∞, 0] nonnegative