Proof methods / preserver_image
Image under a real-rootedness preserver
Parameters (20)
- all
- derivative_polynomial, shift 0 3144 proof steps, 2807 sequences
- euler_polynomial, shift 0 2783 proof steps, 2420 sequences
- affine, shift 0 1772 proof steps, 1584 sequences
- reversal, shift 0 1263 proof steps, 1112 sequences
- mobius, shift 0 1219 proof steps, 1093 sequences
- reversal, shift 1 7 proof steps, 5 sequences
- hadamard_multiplier, shift 0 6 proof steps, 5 sequences
- affine, shift -1 6 proof steps, 5 sequences
- affine, shift 1 5 proof steps, 4 sequences
- hadamard_multiplier, shift 2 4 proof steps, 3 sequences
- euler_polynomial, shift 1 3 proof steps, 2 sequences
- affine, shift 2 3 proof steps, 3 sequences
- derivative_polynomial, shift 2 2 proof steps, 1 sequences
- euler_polynomial, shift 2 2 proof steps, 1 sequences
- hadamard_multiplier, shift -1 2 proof steps, 2 sequences
- derivative_of_power_multiple, shift 1 2 proof steps, 1 sequences
- derivative_polynomial, shift -1 1 proof steps, 1 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
8,704 sequences
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A038305 → basis A013610 → x ↦ x/(x+1)
169, 572, 484 · 2197, 11154, 18876, 10648 … Σ 1225, 42875, 1500625, 52521875, …
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A156992 → basis A008277 → x ↦ x/(x+1)
1 · 4, 6 · 24, 78, 60 · 192, 1032, 1704, 888 … Σ 1, 10, 162, 3816, 123000, 5199120, …
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A038314 → basis A013609 → D
1056, 1152 · 34848, 76032, 41472 … Σ 2208, 152352, 9344256, 537294720, …
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A078812 → basis A013611 → reverse
1 · 4, 3 · 16, 24, 8 · 64, 144, 100, 21 … Σ 1, 7, 48, 329, 2255, 15456, 105937, …
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A257615 → (θ+1)²
1 · 5, 20 · 25, 280, 225 · 125, 2860, 6435, 2000 … Σ 1, 25, 530, 11420, 259680, 6296640, …
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A120910 → basis A013611 → θ+1
3 · 9, 24 · 27, 144, 144 · 81, 648, 1296, 768 … Σ 3, 33, 315, 2793, 23667, 194481, …
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A136572 → (θ+1)² → basis A007318
1 · 4, 4 · 18, 36, 18 · 96, 288, 288, 96 … Σ 1, 8, 72, 768, 9600, 138240, 2257920, …
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A038333 → basis A013609 → x ↦ x/(x+1)
361, 1254, 1089 · 6859, 35739, 62073, 35937 … Σ 2704, 140608, 7311616, 380204032, …
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A091042 → basis A013609 → θ+1
1 · 4, 12 · 16, 80, 60 · 64, 448, 672, 224 … Σ 1, 16, 156, 1408, 12304, 105216, …
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A269947 → θ+1
1 · 0, 2 · 0, 2, 3 · 0, 16, 27, 4 … Σ 1, 2, 5, 47, 1334, 87214, 11021724, …
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A232535 → basis A007318 → D
2 · 14, 6 · 80, 74, 12 · 416, 592, 228, 20 … Σ 0, 2, 20, 166, 1256, 9002, 62268, …
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A085841 → x ↦ x/(x+1)
1 · 3, 7 · 5, 50, 61 · 7, 161, 637, 547 … Σ 1, 10, 116, 1352, 15760, 183712, …
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A120986 → D
1 · 6, 2 · 28, 24, 3 · 120, 180, 60, 4 … Σ 0, 1, 8, 55, 364, 2380, 15504, 100947, …
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A091044 → basis A013610 → x ↦ x/(x+1)
1 · 4, 10 · 16, 80, 91 · 64, 480, 1128, 820 … Σ 1, 14, 187, 2492, 33205, 442442, …
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A225471 → basis A007318 → (θ+1)²
1 · 4, 4 · 32, 48, 9 · 384, 704, 216, 16 … Σ 1, 8, 89, 1320, 24649, 556352, …
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A257627 → x ↦ x/(x+1)
1 · 3, 6 · 9, 78, 78 · 27, 834, 2340, 1560 … Σ 1, 9, 165, 4761, 185565, 9103329, …
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A131689 → θ+1 → reverse
1 · 2 · 6, 2 · 24, 18, 2 … Σ 1, 2, 8, 44, 308, 2612, 25988, 296564, …
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A159854 → x ↦ x+1 → (θ+1)²
1, 8, 9 · 2, 20, 36, 16 · 4, 48, 117, 96, 25 … Σ 18, 74, 290, 1098, 4050, 14634, 52002, …
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A290596 → x ↦ x+1
1 · 3, 1 · 21, 12, 1 · 225, 171, 27, 1 … Σ 1, 4, 34, 424, 6880, 136672, 3202048, …
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A132393 → D+1 → x ↦ x/(x+1)
1 · 1, 2 · 1, 5, 5 · 2, 14, 28, 17 … Σ 1, 3, 11, 61, 457, 4323, 49443, 663549, …
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A038208 → basis A013611 → D
8 · 64, 128 · 384, 1536, 1536 … Σ 0, 8, 192, 3456, 55296, 829440, …
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basis-map coefficients of A371080, shifted → x ↦ x/(x+1)
1 · 0, 1 · 0, 3, 4 · 0, 18, 45, 28 … Σ 1, 1, 7, 91, 1729, 43225, 1339975, …
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basis-map coefficients of A112227 → basis A013611 → (θ+1)²
1, 32, 144 · 1, 48, 432, 1024 … Σ 177, 1505, 11425, 80625, 540625, …
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A225471 → D
1 · 10, 2 · 131, 42, 3 · 2196, 892, 108, 4 … Σ 0, 1, 12, 176, 3200, 70144, 1806336, …
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A122542 → D
1 · 2, 2 · 2, 8, 3 · 2, 16, 18, 4 … Σ 0, 1, 4, 13, 40, 117, 332, 921, …
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A062145 → x ↦ x+1 → x ↦ x/(x+1)
1 · 5, 9 · 21, 72, 61 · 84, 420, 693, 377 … Σ 1, 14, 154, 1574, 15619, 152900, …
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A059344 → basis A013610 → D+1
1 · 1 · 9, 6 · 25, 18 … Σ 1, 1, 15, 43, 565, 2541, 32371, 191815, …
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A078812 → D → θ+1
1 · 4, 4 · 10, 24, 9 · 20, 84, 72, 16 … Σ 0, 1, 8, 43, 192, 768, 2856, 10085, …
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A059364 → basis A007318 → basis A013609
1 · 4, 2 · 28, 30, 8 · 280, 468, 260, 48 … Σ 1, 6, 66, 1056, 22176, 576576, …
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basis-map coefficients of A026729 → (θ+1)² → D
4 · 0, 18 · 0, 0, 48 · 0, 0, 0, 100 … Σ 0, 4, 18, 48, 100, 180, 294, 448, …
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A038338 → basis A013609 → basis A007318
2304, 2304, 576 · 110592, 165888, 82944, 13824 … Σ 5184, 373248, 26873856, 1934917632, …
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A154602 → x ↦ x+1
1 · 2, 1 · 8, 6, 1 · 40, 40, 12, 1 … Σ 1, 3, 15, 93, 681, 5691, 53079, 544053, …
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A063967 → basis A013609 → D
2 · 10, 8 · 40, 64, 24 · 144, 344, 264, 64 … Σ 0, 2, 18, 128, 816, 4896, 28256, …
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A144445 → θ+1
1 · 1, 2 · 1, 46, 3 · 1, 412, 618, 4 … Σ 1, 3, 50, 1035, 29091, 1028328, …
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A011973 → D → D+1
1 · 2 · 5, 2 · 10, 6 … Σ 0, 0, 1, 2, 7, 16, 38, 82, …
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basis-map coefficients of A026729, shifted → (θ+1)² → D
-4 · 0, 18 · 0, 0, -48 · 0, 0, 0, 100 … Σ 0, -4, 18, -48, 100, -180, 294, -448, …
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A228955 → D+1
1 · 2, 1 · 8, 6 · 42, 48, 6 … Σ 1, 3, 14, 96, 864, 9600, 126720, …
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A121574 → x ↦ x+1 → θ+1
1 · 3, 2 · 10, 14, 3 · 33, 70, 33, 4 … Σ 1, 5, 27, 140, 712, 3564, 17620, 86240, …
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A397048 → x ↦ x/(x+1)
1 · 1, 2 · 4, 13, 10 · 27, 118, 167, 77 … Σ 1, 3, 27, 389, 7697, 193707, 5918139, …
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A330510 → x ↦ x/(x+1)
2 · 3, 6 · 4, 18, 18 · 5, 37, 81, 54 … Σ 2, 9, 40, 177, 782, 3453, 15244, 67293, …
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A003506 → D
2 · 6, 6 · 12, 24, 12 · 20, 60, 60, 20 … Σ 0, 0, 2, 12, 48, 160, 480, 1344, …
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A034870 → (θ+1)²
1 · 1, 8, 9 · 1, 16, 54, 64, 25 … Σ 1, 18, 160, 1120, 6912, 39424, 212992, …
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A200536 → (θ+1)²
1 · 1, 12, 18 · 1, 24, 117, 192, 100 … Σ 1, 31, 434, 4680, 44064, 381456, …
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A008277 → (θ+1)² → reverse
1 · 4, 1 · 9, 12, 1 · 16, 54, 28, 1 … Σ 1, 5, 22, 99, 471, 2386, 12867, 73681, …
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A145597 → basis A007318
1 · 6, 3 · 27, 27, 6 · 110, 165, 75, 10 … Σ 1, 9, 60, 360, 2061, 11529, 63728, …
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A114655 → reverse
2 · 2, 4 · 2, 12, 8 · 2, 24, 48, 16 … Σ 2, 6, 22, 90, 394, 1806, 8558, 41586, …
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A103633 → D+1
1 · 1, 1 · 1, 3, 1 · 0, 2, 4, 1 … Σ 1, 2, 5, 7, 16, 20, 44, 52, …
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A274498 → x ↦ x+1 → D
6 · 18 · 108, 72 · 324, 216 … Σ 0, 0, 6, 18, 180, 540, 4050, 12150, …
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A060920 → Veronese 2 (j=1) → x ↦ x+1
1 · 5 · 21, 1 · 85, 14 … Σ 0, 1, 5, 22, 99, 449, 2040, 9273, …
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A059473 → x ↦ x+1
1 · 4, 2 · 18, 18, 4 · 80, 120, 56, 8 … Σ 1, 6, 40, 264, 1744, 11520, 76096, …