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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Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
144, 240, 100 · 1728, 4320, 3600, 1000 … Σ 484, 10648, 234256, 5153632, 113379904, …
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A038301 → reverse → D
198, 162 · 3267, 5346, 2187 … Σ 360, 10800, 288000, 7200000, 172800000, …
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A038323 → basis A013610 → x ↦ x/(x+1)
1 · 20, 47 · 400, 1880, 2209 … Σ 1, 67, 4489, 300763, 20151121, …
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A038221 → D+1
1 · 6, 3 · 27, 36, 9 · 108, 243, 162, 27 … Σ 1, 9, 72, 540, 3888, 27216, 186624, …
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A144442 → D+1
1 · 2, 1 · 18, 19, 1 · 119, 354, 121, 1 … Σ 1, 3, 38, 595, 12423, 323022, 10050356, …
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A367270 → basis A013609 → θ+1
1 · 1 · 3, 8 · 10, 48, 36 … Σ 1, 1, 11, 94, 763, 6086, 48094, 377644, …
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A207607 → (θ+1)²
1 · 1, 8 · 1, 20, 18 · 1, 36, 81, 32 … Σ 1, 9, 39, 150, 540, 1857, 6173, 19989, …
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A038328 → x ↦ x+1 → reverse
4, 56, 196 · 8, 168, 1176, 2744 … Σ 256, 4096, 65536, 1048576, 16777216, …
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A136124 → basis A013611 → basis A007318
1 · 7, 4 · 56, 60, 16 · 504, 764, 384, 64 … Σ 1, 11, 132, 1716, 24024, 360360, …
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A099040 → θ+1 → reverse
1 · 4 · 12, 4 · 32, 24 … Σ 1, 4, 16, 56, 188, 608, 1920, 5952, …
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A038291 → basis A013609 → θ+1
1 · 10, 4 · 100, 80, 12 · 1000, 1200, 360, 32 … Σ 1, 14, 192, 2592, 34560, 456192, …
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A008459 → D → D
2 · 18, 6 · 72, 96, 12 · 200, 600, 300, 20 … Σ 0, 0, 2, 24, 180, 1120, 6300, 33264, …
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A008292 → (θ+1)²
1 · 1, 4 · 1, 16, 9 · 1, 44, 99, 16 … Σ 1, 5, 26, 160, 1140, 9240, 84000, …
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A113413 → basis A013610 → x ↦ x+1
1 · 6, 3 · 34, 36, 9 · 194, 312, 162, 27 … Σ 1, 9, 79, 695, 6113, 53769, 472943, …
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A122542 → reverse
1 · 1 · 1, 2 · 1, 4, 2 … Σ 1, 1, 3, 7, 17, 41, 99, 239, …
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A204420 → reverse
1 · 1 · 3, 6 · 15, 90, 120 … Σ 1, 1, 9, 225, 11025, 893025, 108056025, …
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A052553 → (θ+1)²
1 · 1 · 1, 4 · 1, 8 … Σ 1, 1, 5, 9, 22, 44, 91, 179, …
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A391045 → reverse → x ↦ x+1
1 · 2, 1 · 6, 6, 1 · 21, 31, 12, 1 … Σ 1, 3, 13, 65, 353, 2023, 12041, 73713, …
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A178822 → x ↦ x/(x+1)
1 · 6, 12 · 21, 84, 84 · 56, 336, 672, 448 … Σ 1, 18, 189, 1512, 10206, 61236, 336798, …
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A286785 → θ+1
14, 28, 6 · 42, 144, 81, 8 · 132, 660, 660, 176, 10 … Σ 48, 275, 1638, 9996, 62016, 389367, …
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A063967 → basis A013610 → D
3 · 15, 18 · 60, 144, 81 · 216, 774, 891, 324 … Σ 0, 3, 33, 285, 2205, 16050, 112350, …
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A094816 → basis A013611 → basis A007318
1 · 6, 4 · 41, 52, 16 · 316, 572, 336, 64 … Σ 1, 10, 109, 1288, 16417, 224686, …
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A130534 → D+1 → θ+1
1 · 2, 2 · 5, 10, 3 · 17, 46, 27, 4 … Σ 1, 4, 18, 94, 570, 3968, 31304, 276568, …
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A132393 → reverse → θ+1
1 · 1 · 1, 2 · 1, 6, 6 … Σ 1, 1, 3, 13, 70, 446, 3276, 27252, …
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A197208 → x ↦ x+1
180, 180, 30 · 1320, 1980, 780, 60 … Σ 390, 4140, 43785, 464184, 4941468, …
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A147716 → basis A013610 → (θ+1)²
1 · 15, 12 · 225, 360, 81 · 3375, 8100, 3645, 432 … Σ 1, 27, 666, 15552, 349920, 7663248, …
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A060187 → x ↦ x/(x+1)
1 · 1, 2 · 1, 8, 8 · 1, 26, 72, 48 … Σ 1, 3, 17, 147, 1697, 24483, 423857, …
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A075500 → θ+1
25, 30, 3 · 125, 350, 90, 4 · 625, 3750, 1875, 200, 5 … Σ 58, 569, 6455, 82506, 1166887, …
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A069466 → γ → D+1
1 · 2 · 18, 12 · 140, 120 … Σ 1, 2, 30, 260, 3010, 33012, 379764, …
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A290598 → θ+1
1 · 4, 2 · 28, 28, 3 · 280, 420, 90, 4 … Σ 1, 6, 59, 794, 13473, 274966, 6544291, …
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A069466 → basis A007318
1 · 4, 2 · 36, 36, 6 · 400, 600, 240, 20 … Σ 1, 6, 78, 1260, 22470, 424116, 8305836, …
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basis-map coefficients of Triangle read by rows: row n consists of n-1 zeros followed by Fibonacci(n).
1 · 1, 1 · 2, 4, 2 · 3, 9, 9, 3 … Σ 1, 2, 8, 24, 80, 256, 832, 2688, …
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A133087 → reverse → x ↦ x+1
1 · 3, 2 · 4, 4, 1 · 12, 20, 11, 2 … Σ 1, 5, 9, 45, 81, 405, 729, 3645, …
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A091042 → reverse → (θ+1)²
1 · 3, 4 · 5, 40, 9 · 7, 140, 189, 16 … Σ 1, 7, 54, 352, 2080, 11520, 60928, …
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A081579: Pascal-(1,4,1) array.
1 · 1, 1 · 1, 6, 1 · 1, 11, 11, 1 … Σ 1, 2, 8, 24, 80, 256, 832, 2688, …
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basis-map coefficients of A297191, shifted → D → (θ+1)²
0, 8 · 0, 0, 0, 64 · 0, 0, 0, 0, 0, 216 … Σ 0, 8, 64, 216, 512, 1000, 1728, 2744, …
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A078812 → D → D
2 · 12, 6 · 42, 48, 12 · 112, 216, 120, 20 … Σ 0, 0, 2, 18, 102, 468, 1902, 7146, …
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A038321 → D
154, 98 · 2541, 3234, 1029 … Σ 252, 6804, 163296, 3674160, 79361856, …
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A292219 → x ↦ x+1
1 · 7, 1 · 81, 22, 1 · 1303, 507, 45, 1 … Σ 1, 8, 104, 1856, 41920, 1142272, …
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A125662 → reverse → D+1
1 · 4, 3 · 7, 22, 8 · 10, 59, 88, 21 … Σ 1, 7, 37, 178, 813, 3589, 15465, 65444, …
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A081582 → D+1
1 · 2, 1 · 10, 11, 1 · 18, 51, 20, 1 … Σ 1, 3, 22, 90, 447, 1925, 8572, 36612, …
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A113953 → x ↦ x−6
1 · -6, 1 · 24, -10, 1 · -72, 60, -14, 1 … Σ 1, -5, 15, -25, -25, 375, -1625, 4375, …
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A038231 → θ+1
1 · 4, 2 · 16, 16, 3 · 64, 96, 36, 4 … Σ 1, 6, 35, 200, 1125, 6250, 34375, …
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A282629 → x ↦ x+1
1 · 4, 3 · 25, 33, 9 · 199, 360, 189, 27 … Σ 1, 7, 67, 775, 10339, 154903, 2560339, …
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A142458 → D+1
1 · 2, 1 · 9, 10, 1 · 40, 117, 42, 1 … Σ 1, 3, 20, 200, 2640, 43120, 837760, …
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delannoy → D
1 · 3, 2 · 5, 10, 3 · 7, 26, 21, 4 … Σ 0, 1, 5, 18, 58, 175, 507, 1428, …
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A049374 → x ↦ x+1
1 · 7, 1 · 61, 20, 1 · 649, 351, 39, 1 … Σ 1, 8, 82, 1040, 15760, 277040, 5522560, …
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A034839 → θ+1
1 · 1 · 1, 2 · 1, 6 … Σ 1, 1, 3, 7, 16, 36, 80, 176, …
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A038331 → x ↦ x+1 → x ↦ x/(x+1)
289, 748, 484 · 4913, 19074, 24684, 10648 … Σ 1521, 59319, 2313441, 90224199, …
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A290595 → reverse
1 · 2, 1 · 4, 19, 4 · 8, 147, 222, 28 … Σ 1, 3, 27, 405, 8505, 229635, 7577955, …