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
- affine, shift 2 3 proof steps, 3 sequences
- euler_polynomial, shift 1 3 proof steps, 2 sequences
- derivative_of_power_multiple, shift 1 2 proof steps, 1 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
- reversal, shift -1 1 proof steps, 1 sequences
- derivative_of_power_multiple, shift 0 1 proof steps, 1 sequences
- derivative_polynomial, shift -1 1 proof steps, 1 sequences
- hadamard_multiplier, shift 1 1 proof steps, 1 sequences
8,704 sequences
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A160562 → x ↦ x/(x+1)
1 · 1, 2 · 1, 12, 12 · 1, 94, 220, 128 … Σ 1, 3, 25, 443, 10609, 338547, 14498185, …
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A013622 → basis A013610 → D
15 · 240, 450 · 2880, 10800, 10125 … Σ 0, 15, 690, 23805, 730020, 20988075, …
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A114687 → θ+1 → x ↦ x+1
1 · 5, 4 · 25, 36, 12 · 129, 264, 168, 32 … Σ 1, 9, 73, 593, 4881, 40665, 342105, …
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A169803 → basis A013610 → D+1
1 · 5, 3 · 9, 6 · 20, 33, 9 … Σ 1, 8, 15, 62, 149, 484, 1275, 3754, …
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A038330 → D
96, 32 · 1728, 1152, 192 · 27648, 27648, 9216, 1024 … Σ 128, 3072, 65536, 1310720, 25165824, …
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A094816 → D → θ+1
1 · 3, 4 · 8, 24, 9 · 24, 116, 90, 16 … Σ 0, 1, 7, 41, 246, 1609, 11697, 94737, …
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A008292 → θ+1 (shift +2)
1, 8, 3 · 1, 22, 33, 4 · 1, 52, 198, 104, 5 … Σ 12, 60, 360, 2520, 20160, 181440, …
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A145324 → basis A013611 → reverse
1 · 8, 3 · 96, 68, 12 · 1536, 1568, 532, 60 … Σ 1, 11, 176, 3696, 96096, 2978976, …
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A125092 → D+1 → (θ+1)²
1 · 5, 16 · 9, 104, 81 · 13, 264, 675, 256 … Σ 1, 21, 194, 1208, 5976, 25432, 97312, …
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A038338 → basis A013610 → θ+1
576, 3456, 3888 · 13824, 124416, 279936, 186624 … Σ 7920, 604800, 44064000, 3110400000, …
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A008292 → θ+1 (shift +2) → x ↦ x/(x+1)
1, 10, 12 · 1, 25, 80, 60 · 1, 56, 360, 660, 360 … Σ 23, 166, 1437, 14512, 167491, 2174746, …
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A054142 → Veronese 2 (j=1) → D+1
1 · 3 · 6, 1 · 17, 10 … Σ 0, 1, 3, 7, 27, 82, 242, 725, …
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basis-map coefficients of A030528, shifted → basis A013609 → (θ+1)²
1 · -1, -8 · 1, 16, 36 · -1, -24, -108, -128 … Σ 1, -9, 53, -261, 1161, -4833, 19197, …
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A133214 → basis A013609 → reverse
16, 32, 13 · 64, 240, 228, 63 … Σ 61, 595, 6145, 65527, 712909, 7863667, …
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A216973 → θ+1 → D
4 · 12, 18 · 24, 72, 48 · 40, 180, 240, 100 … Σ 0, 0, 4, 30, 144, 560, 1920, 6048, …
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A062196 → D
3 · 8, 12 · 15, 60, 30 · 24, 180, 240, 60 … Σ 0, 3, 20, 105, 504, 2310, 10296, 45045, …
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basis-map coefficients of A112227 → basis A008277 → reverse
1, 3, 1 · 1, 6, 7, 1 · 1, 10, 25, 15, 1 … Σ 5, 15, 52, 203, 877, 4140, 21147, …
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A038277 → reverse
1 · 11, 7 · 121, 154, 49 · 1331, 2541, 1617, 343 … Σ 1, 18, 324, 5832, 104976, 1889568, …
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A093375 → Veronese 2 → D
1 · 6 · 18, 2 · 40, 20 … Σ 0, 0, 1, 6, 20, 60, 168, 448, …
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A143497 → reverse
1, 10, 20 · 1, 18, 90, 120 · 1, 28, 252, 840, 840 … Σ 31, 229, 1961, 19081, 207775, 2501801, …
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A196347 → basis A013610 → (θ+1)²
1 · 2, 12 · 8, 96, 162 · 48, 864, 2916, 2592 … Σ 1, 14, 266, 6420, 187800, 6450000, …
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A038221 → basis A013609 → (θ+1)²
1 · 6, 24 · 36, 288, 324 · 216, 2592, 5832, 3456 … Σ 1, 30, 648, 12096, 207360, 3359232, …
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A038322 → (θ+1)²
1 · 11, 32 · 121, 704, 576 · 1331, 11616, 19008, 8192 … Σ 1, 43, 1401, 40147, 1066033, 26894139, …
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A272866 → Veronese 2 → D
1 · 11, 2 · 30, 60, 3 · 58, 390, 174, 4 … Σ 0, 1, 13, 93, 626, 3905, 23439, 136717, …
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A056939 → D+1
1 · 2, 1 · 5, 6, 1 · 11, 30, 13, 1 … Σ 1, 3, 12, 55, 276, 1477, 8296, 48393, …
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basis-map coefficients of A113214, shifted → Veronese 2 (j=1) → basis A007318
-1 · 2 · -4, -1 · 8, 4 … Σ 0, -1, 2, -5, 12, -29, 70, -169, …
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A038281 → x ↦ x+1 → reverse
9, 66, 121 · 27, 297, 1089, 1331 … Σ 196, 2744, 38416, 537824, 7529536, …
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Hadamard-factor parent of A127773 → D+1
1 · 3, 3 · 0, 24, 12 · 0, 0, 180, 60 … Σ 1, 6, 36, 240, 1800, 15120, 141120, …
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A089627 → x ↦ x+1 → D+1
1 · 1 · 5, 2 · 13, 6 … Σ 1, 1, 7, 19, 85, 301, 1211, 4607, …
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A003506 → (θ+1)² → D
8 · 24, 54 · 48, 216, 192 · 80, 540, 960, 500 … Σ 0, 0, 8, 78, 456, 2080, 8160, 28896, …
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A367270 → basis A013610 → (θ+1)²
1 · 1 · 3, 24 · 10, 144, 243 … Σ 1, 1, 27, 397, 4913, 56541, 624675, …
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A066325 → D+1
1 · 1, 1 · -1, 2, 1 · -3, -3, 3, 1 … Σ 1, 2, 2, -2, -10, -4, 52, 92, …
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A109692 → basis A013609 → reverse
1 · 2, 2 · 12, 20, 8 · 120, 272, 200, 48 … Σ 1, 4, 40, 640, 14080, 394240, 13404160, …
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A125092 → D+1
1 · 5, 4 · 9, 26, 9 · 13, 66, 75, 16 … Σ 1, 9, 44, 170, 576, 1792, 5248, 14688, …
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A110441 → basis A013609 → (θ+1)²
1 · 4, 8 · 14, 64, 36 · 48, 352, 432, 128 … Σ 1, 12, 114, 960, 7516, 56016, 402808, …
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A038208 → Veronese 2 (j=1) → basis A007318
2 · 8 · 32, 8 · 128, 64 … Σ 0, 2, 8, 40, 192, 928, 4480, 21632, …
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A169803 → basis A013610 → θ+1
1 · 2, 6 · 3, 12 · 5, 30, 27 … Σ 1, 8, 15, 62, 149, 484, 1275, 3754, …
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A099093 → basis A013609 → reverse
1 · 6, 3 · 36, 42, 12 · 216, 396, 234, 45 … Σ 1, 9, 90, 891, 8829, 87480, 866781, …
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A166341 → x ↦ x/(x+1)
1, 12, 12 · 1, 26, 72, 48 · 1, 54, 294, 480, 240 … Σ 25, 147, 1069, 9291, 94045, 1086987, …
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A013612 → θ+1
1, 20, 75 · 1, 30, 225, 500 · 1, 40, 450, 2000, 3125 … Σ 96, 756, 5616, 40176, 279936, 1912896, …
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A133087 → D
1 · 2, 2 · 5, 8, 3 · 4, 12, 12, 4 … Σ 0, 1, 4, 16, 32, 112, 192, 640, …
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A116414 → reverse → x ↦ x+1
1 · 5, 4 · 22, 34, 13 · 95, 216, 162, 40 … Σ 1, 9, 69, 513, 3789, 27945, 206037, …
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A008955 → basis A013610 → x ↦ x/(x+1)
1 · 2, 5 · 10, 59, 85 · 100, 960, 3033, 3145 … Σ 1, 7, 154, 7238, 593516, 75376532, …
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A030528 → θ+1
1 · 1, 2 · 0, 4, 3 · 0, 2, 9, 4 … Σ 1, 3, 7, 15, 30, 58, 109, 201, …
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A011973 → D → (θ+1)²
1 · 2 · 3, 8 · 4, 24 … Σ 0, 0, 1, 2, 11, 28, 80, 194, …
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A249251 → basis A007318 → x ↦ x/(x+1)
1 · 4, 7 · 13, 44, 37 · 43, 216, 360, 199 … Σ 1, 11, 94, 818, 7108, 61772, 536824, …
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A104684 → basis A013609 → reverse
1 · 2, 3 · 4, 16, 13 · 8, 60, 114, 63 … Σ 1, 5, 33, 245, 1921, 15525, 127905, …
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A038319 → basis A013610 → x ↦ x/(x+1)
256, 992, 961 · 4096, 23808, 46128, 29791 … Σ 2209, 103823, 4879681, 229345007, …
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A225468 → x ↦ x+1
1 · 3, 1 · 12, 9, 1 · 63, 72, 18, 1 … Σ 1, 4, 22, 154, 1282, 12226, 130690, …
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A178820 → Veronese 2 → D+1
1 · 4 · 20, 10 · 80, 60 … Σ 1, 4, 30, 140, 560, 2016, 6720, 21120, …