preserver_image / euler_polynomial, shift 0

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352 sequences

• A008278 → x ↦ x+1 (shift +1) → θ+1
  1 · 2, 2 · 5, 10, 3 · 15, 46, 30, 4 … row sums 1, 4, 18, 95, 579, 3966, 29944, 246141, …
  (−∞, 0] nonnegative
• A059344 → (θ+1)²
  1 · 1 · 1, 8 · 1, 24 … row sums 1, 1, 9, 25, 157, 621, 3661, 17389, …
  (−∞, 0] nonnegative
• A099093 → (θ+1)²
  1 · 0, 12 · 0, 12, 81 · 0, 0, 162, 432 … row sums 1, 12, 93, 594, 3402, 18144, 92043, …
  (−∞, 0] nonnegative
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
  1, 4, 3 · 3, 18, 27, 12 · 6, 48, 108, 96, 30 … row sums 0, 0, 8, 60, 288, 1120, 3840, 12096, …
  (−∞, 0] nonnegative rational roots
• A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
  A007318 → (θ+1)²
  1 · 1, 4 · 1, 8, 9 · 1, 12, 27, 16 … row sums 1, 5, 18, 56, 160, 432, 1120, 2816, …
  A125092 (−∞, 0] nonnegative
• A113413 → (θ+1)²
  1 · 2, 4 · 2, 16, 9 · 2, 32, 54, 16 … row sums 1, 6, 27, 104, 365, 1206, 3815, 11680, …
  (−∞, 0] nonnegative
• A130123 → θ+1
  1 · 0, 4 · 0, 0, 12 · 0, 0, 0, 32 … row sums 1, 4, 12, 32, 80, 192, 448, 1024, …
  (−∞, 0] nonnegative rational roots
• basis-map coefficients of A026729, shifted → θ+1
  1 · 0, -2 · 0, 0, 3 · 0, 0, 0, -4 … row sums 1, -2, 3, -4, 5, -6, 7, -8, …
  (−∞, 0] rational roots
• A013620 → θ+1
  1 · 2, 6 · 4, 24, 27 · 8, 72, 162, 108 … row sums 1, 8, 55, 350, 2125, 12500, 71875, …
  (−∞, 0] nonnegative rational roots
• A023531 → (θ+1)² → θ+1
  1 · 0, 8 · 0, 0, 27 · 0, 0, 0, 64 … row sums 1, 8, 27, 64, 125, 216, 343, 512, …
  (−∞, 0] nonnegative rational roots
• A081579 → θ+1
  1 · 1, 2 · 1, 12, 3 · 1, 22, 33, 4 … row sums 1, 3, 16, 60, 240, 896, 3328, 12096, …
  (−∞, 0] nonnegative
• A039756 → (θ+1)²
  1 · 1, 4 · 1, 16, 9 · 1, 36, 117, 16 … row sums 1, 5, 26, 170, 1252, 9972, 85448, …
  (−∞, 0] nonnegative
• A011973 → x ↦ x+2 (shift +1) → θ+1
  1 · 3, 2 · 5, 4 · 11, 14, 3 … row sums 1, 5, 9, 28, 62, 165, 391, 983, …
  (−∞, 0] nonnegative
• A207607 → (θ+1)²
  1 · 1, 8 · 1, 20, 18 · 1, 36, 81, 32 … row sums 1, 9, 39, 150, 540, 1857, 6173, 19989, …
  (−∞, 0] nonnegative
• A008292 → (θ+1)²
  1 · 1, 4 · 1, 16, 9 · 1, 44, 99, 16 … row sums 1, 5, 26, 160, 1140, 9240, 84000, …
  (−∞, 0] nonnegative
• A052553 → (θ+1)²
  1 · 1 · 1, 4 · 1, 8 … row sums 1, 1, 5, 9, 22, 44, 91, 179, …
  (−∞, 0] nonnegative
• A038231 → θ+1
  1 · 4, 2 · 16, 16, 3 · 64, 96, 36, 4 … row sums 1, 6, 35, 200, 1125, 6250, 34375, …
  (−∞, 0] nonnegative rational roots
• A099040 → (θ+1)²
  1 · 0, 8 · 0, 8, 36 · 0, 0, 72, 128 … row sums 1, 8, 44, 200, 820, 3136, 11424, 40128, …
  (−∞, 0] nonnegative
• A113413 → θ+1
  1 · 2, 2 · 2, 8, 3 · 2, 16, 18, 4 … row sums 1, 4, 13, 40, 117, 332, 921, 2512, …
  (−∞, 0] nonnegative
• basis-map coefficients of A112227 → (θ+1)²
  0, 0, 9 · 0, 0, 0, 16 · 0, 0, 0, 0, 25 … row sums 9, 16, 25, 36, 49, 64, 81, 100, …
  (−∞, 0] nonnegative rational roots
• A007318 → (1/3+(1/2)n+(1/6)n²)θ+1+(3/2)n+(1/2)n² → (θ+1)²
  1 · 3, 16 · 6, 64, 90 · 10, 160, 450, 320 … row sums 1, 19, 160, 940, 4480, 18592, 69888, …
  (−∞, 0] nonnegative
• A007318 → (1/2+(1/2)n)θ+1+n → θ+1
  1 · 2, 6 · 3, 18, 18 · 4, 36, 72, 40 … row sums 1, 8, 39, 152, 520, 1632, 4816, 13568, …
  (−∞, 0] nonnegative
• A249252 → θ+1
  1 · 1, 8 · 1, 16, 24 · 1, 24, 96, 64 … row sums 1, 9, 41, 185, 793, 3305, 13481, 54105, …
  (−∞, 0] nonnegative
• A081579 → (θ+1)²
  1 · 1, 4 · 1, 24, 9 · 1, 44, 99, 16 … row sums 1, 5, 34, 160, 760, 3296, 13920, 56704, …
  (−∞, 0] nonnegative
• A104684 → (θ+1)²
  1 · 2, 4 · 6, 24, 9 · 20, 120, 108, 16 … row sums 1, 6, 39, 264, 1785, 11958, 79303, …
  (−∞, 0] nonnegative
• A113953 → (θ+1)²
  1 · 0, 4 · 0, 8, 9 · 0, 0, 36, 16 … row sums 1, 4, 17, 52, 157, 428, 1137, 2892, …
  (−∞, 0] nonnegative
• A169803 → (θ+1)²
  1 · 1, 4 · 1, 8 · 1, 12, 9 … row sums 1, 5, 9, 22, 44, 91, 179, 349, …
  (−∞, 0] nonnegative
• A321331: Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).
  A008277 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 14, 18, 4 … row sums 1, 3, 10, 37, 151, 674, 3263, 17007, …
  A321331 (−∞, 0] nonnegative
• A249248 → (θ+1)²
  1 · 1, 12 · 1, 28, 27 · 1, 48, 198, 48 … row sums 1, 13, 56, 295, 1409, 7021, 34808, …
  (−∞, 0] nonnegative
• basis-map coefficients of A297191, shifted → θ+1
  1 · 0, 0, 3 · 0, 0, 0, 0, 5 · 0, 0, 0, 0, 0, 0, 7 … row sums 1, 3, 5, 7, 9, 11, 13, 15, …
  (−∞, 0] nonnegative rational roots
• A013609 → θ+1
  1 · 1, 4 · 1, 8, 12 · 1, 12, 36, 32 … row sums 1, 5, 21, 81, 297, 1053, 3645, 12393, …
  (−∞, 0] nonnegative rational roots
• A130534 → (θ+1)²
  1 · 1, 4 · 2, 12, 9 · 6, 44, 54, 16 … row sums 1, 5, 23, 120, 724, 5012, 39332, …
  (−∞, 0] nonnegative
• basis-map coefficients of A297191, shifted → (θ+1)²
  1 · 0, 0, 9 · 0, 0, 0, 0, 25 · 0, 0, 0, 0, 0, 0, 49 … row sums 1, 9, 25, 49, 81, 121, 169, 225, …
  (−∞, 0] nonnegative rational roots
• A249250 → θ+1
  1 · 1, 6 · 1, 12, 9 · 1, 18, 45, 12 … row sums 1, 7, 22, 76, 244, 772, 2392, 7312, …
  (−∞, 0] nonnegative
• A011973 → x ↦ x−1/4 → (θ+1)²
  1 · 2 · 3, 16 · 4, 64 … row sums 1, 2, 19, 68, 309, 1190, 4615, 17160, …
  (−∞, 0] nonnegative
• A120910 → (θ+1)²
  3 · 6, 12 · 12, 48, 27 · 24, 144, 162, 48 … row sums 3, 18, 87, 378, 1539, 5994, 22599, …
  (−∞, 0] nonnegative
• A116412 → (θ+1)²
  1 · 3, 4 · 6, 24, 9 · 12, 84, 81, 16 … row sums 1, 7, 39, 193, 886, 3870, 16303, 66811, …
  (−∞, 0] nonnegative
• A113953 → θ+1
  1 · 0, 2 · 0, 4, 3 · 0, 0, 12, 4 … row sums 1, 2, 7, 16, 41, 94, 219, 492, …
  (−∞, 0] nonnegative
• A091042 → (θ+1)²
  1 · 1, 12 · 1, 40, 45 · 1, 84, 315, 112 … row sums 1, 13, 86, 512, 2848, 15104, 77312, …
  (−∞, 0] nonnegative
• A026374 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 8, 12, 4 … row sums 1, 3, 10, 25, 75, 175, 500, 1125, …
  (−∞, 0] nonnegative
• A007318 → D+1 → (θ+1)²
  1 · 2, 4 · 3, 16, 9 · 4, 36, 54, 16 … row sums 1, 6, 28, 110, 384, 1232, 3712, 10656, …
  (−∞, 0] nonnegative
• A027465 → (θ+1)²
  1 · 3, 4 · 9, 24, 9 · 27, 108, 81, 16 … row sums 1, 7, 42, 232, 1216, 6144, 30208, …
  (−∞, 0] nonnegative
• A023531 → 2D+1 → θ+1
  1 · 2, 2 · 0, 8, 3 · 0, 0, 18, 4 … row sums 1, 4, 11, 22, 37, 56, 79, 106, …
  (−∞, 0] nonnegative rational roots
• A133087 → (θ+1)²
  1 · 2, 4 · 1, 8, 9 · 2, 20, 36, 16 … row sums 1, 6, 18, 74, 160, 592, 1120, 3936, …
  (−∞, 0] nonnegative
• A011973 → θ+1
  1 · 1 · 1, 2 · 1, 4 … row sums 1, 1, 3, 5, 10, 18, 33, 59, …
  (−∞, 0] nonnegative
• A059344 → θ+1
  1 · 1 · 1, 4 · 1, 12 … row sums 1, 1, 5, 13, 61, 221, 1081, 4705, …
  (−∞, 0] nonnegative
• A007318 → θ+1+n → θ+1
  1 · 2, 6 · 3, 16, 15 · 4, 30, 54, 28 … row sums 1, 8, 34, 116, 352, 992, 2656, 6848, …
  (−∞, 0] nonnegative
• A106800 → (θ+1)²
  1 · 1 · 1, 4 · 1, 12, 9 … row sums 1, 1, 5, 22, 104, 531, 2897, 16787, …
  (−∞, 0] nonnegative
• A272866 → (θ+1)²
  1 · 1, 12, 9 · 1, 24, 99, 96, 25 … row sums 1, 22, 245, 2150, 16625, 118750, …
  (−∞, 0] nonnegative
• A007318 → (1/3+(1/2)n+(1/6)n²)θ+1+(3/2)n+(1/2)n² → θ+1
  1 · 3, 8 · 6, 32, 30 · 10, 80, 150, 80 … row sums 1, 11, 68, 320, 1280, 4592, 15232, …
  (−∞, 0] nonnegative