preserver_image / derivative_polynomial, shift 0
Image under a real-rootedness preserver. All operations
Parameters (20)
- all
- derivative_polynomial, shift 0 422 proof steps, 410 sequences
- euler_polynomial, shift 0 367 proof steps, 352 sequences
- affine, shift 0 333 proof steps, 318 sequences
- reversal, shift 0 169 proof steps, 164 sequences
- mobius, shift 0 160 proof steps, 160 sequences
- affine, shift -1 5 proof steps, 5 sequences
- hadamard_multiplier, shift 0 5 proof steps, 5 sequences
- affine, shift 1 4 proof steps, 4 sequences
- reversal, shift 1 4 proof steps, 4 sequences
- hadamard_multiplier, shift 2 3 proof steps, 3 sequences
- affine, shift 2 2 proof steps, 2 sequences
- euler_polynomial, shift 1 2 proof steps, 2 sequences
- hadamard_multiplier, shift -1 2 proof steps, 2 sequences
- hadamard_multiplier, shift 1 1 proof steps, 1 sequences
- reversal, shift -1 1 proof steps, 1 sequences
- derivative_of_power_multiple, shift 0 1 proof steps, 1 sequences
- derivative_of_power_multiple, shift 1 1 proof steps, 1 sequences
- derivative_polynomial, shift -1 1 proof steps, 1 sequences
- derivative_polynomial, shift 2 1 proof steps, 1 sequences
- euler_polynomial, shift 2 1 proof steps, 1 sequences
410 sequences
-
A007318 → D+1 → D+1
1 · 3, 1 · 7, 6, 1 · 13, 21, 9, 1 … row sums 1, 4, 14, 44, 128, 352, 928, 2368, …
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A099093 → D
3 · 3, 18 · 0, 36, 81 · 0, 18, 243, 324 … row sums 0, 3, 21, 117, 585, 2754, 12474, 54999, …
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A060920 → D+1
1 · 3, 1 · 10, 7, 1 · 33, 38, 12, 1 … row sums 1, 4, 18, 84, 390, 1784, 8042, 35804, …
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A007318 → (θ+1)² → D
4 · 8, 18 · 12, 54, 48 · 16, 108, 192, 100 … row sums 0, 4, 26, 114, 416, 1360, 4128, 11872, …
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A126216 → D
1 · 5, 2 · 21, 18, 3 · 84, 112, 42, 4 … row sums 0, 1, 7, 42, 242, 1375, 7773, 43876, …
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A007318 → θ+1 → D+1
1 · 3, 2 · 5, 10, 3 · 7, 24, 21, 4 … row sums 1, 5, 18, 56, 160, 432, 1120, 2816, …
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A078812 → D+1
1 · 3, 1 · 7, 6, 1 · 14, 22, 9, 1 … row sums 1, 4, 14, 46, 145, 444, 1331, 3926, …
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basis-map coefficients of A026729, shifted → D+1
1 · -1, -1 · 0, 2, 1 · 0, 0, -3, -1 … row sums 1, -2, 3, -4, 5, -6, 7, -8, …
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A122542 → D
1 · 2, 2 · 2, 8, 3 · 2, 16, 18, 4 … row sums 0, 1, 4, 13, 40, 117, 332, 921, …
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A007318 → D → D
2 · 6, 6 · 12, 24, 12 · 20, 60, 60, 20 … row sums 0, 0, 2, 12, 48, 160, 480, 1344, …
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A103633 → D+1
1 · 1, 1 · 1, 3, 1 · 0, 2, 4, 1 … row sums 1, 2, 5, 7, 16, 20, 44, 52, …
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chebyshev_t → D+1
1 · 1, 1 · -1, 4, 2 · -3, -3, 12, 4 … row sums 1, 2, 5, 10, 17, 26, 37, 50, …
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A007318 → ×(1+2n+n²) → D+1
1 · 8, 4 · 27, 36, 9 · 64, 144, 96, 16 … row sums 1, 12, 72, 320, 1200, 4032, 12544, …
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A086645 → D+1
1 · 2, 1 · 7, 8, 1 · 16, 45, 18, 1 … row sums 1, 3, 16, 80, 384, 1792, 8192, 36864, …
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A113214 → D
1 · 3, 2 · 2, 8, 3 · 0, 10, 15, 4 … row sums 0, 1, 5, 13, 29, 60, 118, 225, …
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A099097 → D
3 · 1, 18 · 0, 12, 81 · 0, 2, 81, 324 … row sums 0, 3, 19, 93, 407, 1674, 6618, 25455, …
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A272866 → D+1
1 · 4, 5, 1 · 7, 28, 29, 10, 1 … row sums 1, 10, 75, 500, 3125, 18750, 109375, …
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A007318 → ×(1+(3/2)n+(1/2)n²) → D
3 · 12, 12 · 30, 60, 30 · 60, 180, 180, 60 … row sums 0, 3, 24, 120, 480, 1680, 5376, 16128, …
-
A173018 → reverse (shift +1) → D+1
1 · 1, 1 · 1, 3, 1 · 1, 9, 7, 1 … row sums 1, 2, 5, 18, 84, 480, 3240, 25200, …
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A026729 → D+1
1 · 1, 1 · 1, 3, 1 · 0, 4, 5, 1 … row sums 1, 2, 5, 10, 20, 38, 71, 130, …
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A100861 → D
1 · 3 · 6, 6 · 10, 30 … row sums 0, 0, 1, 3, 12, 40, 150, 546, …
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A207607 → D+1
1 · 3, 2 · 6, 9, 2 · 10, 27, 15, 2 … row sums 1, 5, 17, 54, 166, 499, 1475, 4303, …
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A091042 → D
3 · 10, 10 · 21, 70, 21 · 36, 252, 252, 36 … row sums 0, 3, 20, 112, 576, 2816, 13312, 61440, …
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A026374 → D
1 · 3, 2 · 4, 8, 3 · 6, 22, 18, 4 … row sums 0, 1, 5, 15, 50, 125, 375, 875, …
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A091042 → D+1
1 · 4, 3 · 11, 20, 5 · 22, 91, 56, 7 … row sums 1, 7, 36, 176, 832, 3840, 17408, 77824, …
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A367270 → D
2 · 6, 6 · 12, 36, 12 · 20, 120, 120, 20 … row sums 0, 0, 2, 12, 60, 280, 1260, 5544, …
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A008292 → θ+1
1 · 1, 2 · 1, 8, 3 · 1, 22, 33, 4 … row sums 0, 1, 3, 12, 60, 360, 2520, 20160, …
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A028326 → D+1
2 · 4, 2 · 6, 8, 2 · 8, 18, 12, 2 … row sums 2, 6, 16, 40, 96, 224, 512, 1152, …
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A274496 → D
2 · 4 · 8, 8 · 16, 16 … row sums 0, 0, 2, 4, 16, 32, 96, 192, …
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A144438 → D
1 · 5, 2 · 14, 28, 3 · 33, 178, 99, 4 … row sums 0, 1, 7, 45, 314, 2430, 20883, 198310, …
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A094638 → D+1
1 · 2, 1 · 4, 7, 2 · 7, 28, 29, 6 … row sums 1, 3, 13, 70, 446, 3276, 27252, 253296, …
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A063967 → D
1 · 3, 2 · 7, 10, 3 · 15, 32, 21, 4 … row sums 0, 1, 5, 20, 72, 244, 796, 2528, …
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A099093 → D+1
1 · 3, 3 · 3, 21, 9 · 0, 36, 99, 27 … row sums 1, 6, 33, 162, 756, 3402, 14931, 64314, …
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A178822 → D
6 · 42, 42 · 168, 336, 168 · 504, 1512, 1512, 504 … row sums 0, 6, 84, 672, 4032, 20160, 88704, …
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A152828 → D
1 · 1 · 1 · 2, 2 … row sums 0, 0, 0, 1, 1, 1, 4, 4, …
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A124038 → D
1 · -2, 2 · -2, -4, 3 · 4, -6, -6, 4 … row sums 0, 1, 0, -3, -4, 0, 6, 7, …
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A023531 → 2D+1+2n → D
3 · 4, 10 · 0, 12, 21 · 0, 0, 24, 36 … row sums 0, 3, 14, 33, 60, 95, 138, 189, …
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A130123 → D
2 · 0, 8 · 0, 0, 24 · 0, 0, 0, 64 … row sums 0, 2, 8, 24, 64, 160, 384, 896, …
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A081579 → D+1
1 · 2, 1 · 7, 8, 1 · 12, 33, 14, 1 … row sums 1, 3, 16, 60, 240, 896, 3328, 12096, …
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A100861 → reverse → D+1
1 · 1, 1 · 1, 3, 1 · 0, 6, 6, 1 … row sums 1, 2, 5, 13, 38, 116, 382, 1310, …
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A011973 → basis A007318 → D
1 · 2 · 5, 2 · 10, 6 … row sums 0, 0, 1, 2, 7, 16, 41, 94, …
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A026374 → D+1
1 · 2, 1 · 4, 5, 1 · 5, 12, 7, 1 … row sums 1, 3, 10, 25, 75, 175, 500, 1125, …
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A122431 → D
3, 6, 3 · 1, 6, 9, 4 · 0, 2, 9, 12, 5 … row sums 12, 20, 28, 36, 44, 52, 60, 68, …
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A111125 → D+1
1 · 4, 1 · 10, 7, 1 · 21, 28, 10, 1 … row sums 1, 5, 18, 60, 191, 589, 1775, 5257, …
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A091044 → ⊙ C(n+2,k+1) → D
1 · 5, 2 · 14, 28, 3 · 30, 168, 90, 4 … row sums 0, 1, 7, 45, 292, 1930, 12993, 88893, …
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A081578 → D
1 · 5, 2 · 9, 18, 3 · 13, 66, 39, 4 … row sums 0, 1, 7, 30, 122, 455, 1641, 5740, …
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A145596 → D+1
1 · 4, 2 · 11, 14, 3 · 24, 60, 32, 4 … row sums 1, 6, 28, 120, 495, 2002, 8008, 31824, …
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A011973 → x ↦ x+2 (shift +1) → D
1 · 2 · 7, 2 · 16, 6 … row sums 0, 1, 2, 9, 22, 68, 174, 475, …
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basis-map coefficients of A297191, shifted → D+1
1 · 0, 2, 1 · 0, 0, 0, 4, 1 · 0, 0, 0, 0, 0, 6, 1 … row sums 1, 3, 5, 7, 9, 11, 13, 15, …
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A099040 → D
2 · 2, 8 · 0, 16, 24 · 0, 8, 72, 64 … row sums 0, 2, 10, 40, 144, 488, 1592, 5056, …