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_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
- 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
- reversal, shift -1 1 proof steps, 1 sequences
8,704 sequences
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A327997 → reverse
1 · 1, 3 · 1, 7, 9 · 1, 12, 38, 27 … Σ 1, 4, 17, 78, 393, 2208, 13977, 100026, …
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A391533 → Pₙ + Pₙ₊₁
8, 12, 6, 1 · 8, 20, 18, 7, 1 · 8, 28, 38, 25, 8, 1 … Σ 27, 54, 108, 216, 432, 864, 1728, 3456, …
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A120910 → Veronese 2 (j=1) → basis A007318
3 · 12 · 39, 3 · 120, 24 … Σ 0, 3, 12, 42, 144, 492, 1680, 5736, …
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A125662 → D+1
1 · 4, 1 · 14, 8, 1 · 46, 43, 12, 1 … Σ 1, 5, 23, 102, 441, 1871, 7823, 32332, …
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A028421 → x ↦ x/(x+1)
1 · 1, 3 · 2, 10, 11 · 6, 40, 80, 50 … Σ 1, 4, 23, 176, 1689, 19524, 264207, …
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A207824 → basis A013611 → θ+1
1 · 6, 8 · 35, 96, 48 · 204, 848, 864, 256 … Σ 1, 14, 179, 2172, 25461, 291242, …
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A136157 → θ+1
3 · 1, 6 · 0, 2, 9 · 0, 0, 3, 12 … Σ 3, 7, 11, 15, 19, 23, 27, 31, …
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A391533 → reverse → θ+1
1, 8, 12 · 1, 10, 24, 16 · 1, 12, 39, 48, 20 … Σ 21, 51, 120, 276, 624, 1392, 3072, …
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A075499 → reverse
1, 12, 16 · 1, 24, 112, 64 · 1, 40, 400, 960, 256 … Σ 29, 201, 1657, 15821, 170389, 2032785, …
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A035469 → D+1
1 · 5, 1 · 40, 14, 1 · 440, 208, 27, 1 … Σ 1, 6, 55, 676, 10405, 192266, 4144091, …
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A063967 → basis A007318 → θ+1
1 · 2, 2 · 6, 10, 3 · 16, 40, 24, 4 … Σ 1, 4, 19, 84, 366, 1566, 6615, 27648, …
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A249250 → x ↦ x/(x+1)
1 · 1, 4 · 1, 8, 10 · 1, 12, 36, 28 … Σ 1, 5, 19, 77, 307, 1229, 4915, 19661, …
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A026374 → θ+1 → x ↦ x/(x+1)
1 · 1, 3 · 1, 8, 10 · 1, 11, 31, 25 … Σ 1, 4, 19, 68, 297, 1012, 4235, 14036, …
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A048993 → (θ+1)² → x ↦ x/(x+1)
1 · 0, 4 · 0, 4, 13 · 0, 4, 35, 47 … Σ 1, 4, 17, 86, 501, 3280, 23741, 187738, …
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A257612 → x ↦ x/(x+1)
1 · 2, 4 · 4, 32, 32 · 8, 208, 576, 384 … Σ 1, 6, 68, 1176, 27152, 783456, …
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A121574 → basis A013611 → reverse
1 · 4, 3 · 16, 28, 10 · 64, 176, 140, 33 … Σ 1, 7, 54, 413, 3161, 24192, 185149, …
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A136572 → basis A013611 → D
4 · 16, 64 · 72, 576, 1152 · 384, 4608, 18432, 24576 … Σ 0, 4, 80, 1800, 48000, 1500000, …
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A063007 → basis A013609 → (θ+1)²
1 · 3, 16 · 13, 144, 216 · 63, 1056, 3240, 2560 … Σ 1, 19, 373, 6919, 122761, 2106907, …
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A156366 → reverse
9, 12, 1 · 27, 99, 33, 1 · 81, 702, 594, 78, 1 … Σ 22, 160, 1456, 15904, 202672, 2951680, …
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A193685 → x ↦ x/(x+1)
1 · 5, 6 · 25, 61, 37 · 125, 466, 575, 235 … Σ 1, 11, 123, 1401, 16289, 193707, …
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A257606 → x ↦ x/(x+1)
1 · 4, 8 · 16, 72, 72 · 64, 488, 1080, 720 … Σ 1, 12, 160, 2352, 37840, 661872, …
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A038208 → basis A013611 → (θ+1)²
1 · 4, 32 · 16, 256, 576 · 64, 1536, 6912, 8192 … Σ 1, 36, 848, 16704, 297216, 4948992, …
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A126216 → basis A007318 → (θ+1)²
1 · 3, 4 · 11, 28, 9 · 45, 168, 108, 16 … Σ 1, 7, 48, 337, 2414, 17534, 128578, …
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A027467 → basis A013610 → x ↦ x+1
1 · 19, 3 · 361, 114, 9 · 6859, 3249, 513, 27 … Σ 1, 22, 484, 10648, 234256, 5153632, …
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A249248 → reverse → (θ+1)²
1 · 3, 4 · 3, 28, 9 · 3, 88, 108, 16 … Σ 1, 7, 40, 215, 1097, 5621, 28696, …
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A128099 → basis A013611 → D
8 · 16 · 56, 128 · 128, 384 … Σ 0, 0, 8, 16, 184, 512, 3400, 11248, …
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A110441: Triangular array formed by the Mersenne numbers.
1 · 3, 1 · 7, 6, 1 · 15, 23, 9, 1 … Σ 1, 4, 14, 48, 164, 560, 1912, 6528, …
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A116414 → reverse → D
4 · 8, 26 · 12, 84, 120 · 16, 174, 552, 484 … Σ 0, 4, 34, 216, 1226, 6548, 33648, …
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A334218 → θ+1 → x ↦ x+1
1 · 1 · 4, 4 · 15, 24, 9 … Σ 1, 1, 8, 48, 336, 2840, 28200, 321328, …
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A330798 → D
2 · 15, 18 · 84, 224, 144 · 420, 1800, 2475, 1100 … Σ 0, 2, 33, 452, 5795, 71814, 871605, …
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A038331 → basis A013609 → x ↦ x+1
729, 540, 100 · 19683, 21870, 8100, 1000 … Σ 1369, 50653, 1874161, 69343957, …
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A112007: Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.
1 · 2, 1 · 6, 8, 1 · 24, 58, 22, 1 … Σ 1, 3, 15, 105, 945, 10395, 135135, …
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A038277 → (θ+1)²
1 · 7, 44 · 49, 616, 1089 · 343, 6468, 22869, 21296 … Σ 1, 51, 1754, 50976, 1345248, 33324048, …
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A053123 → x ↦ x+1
1 · -1, -2 · 0, 2, 3 · 1, 2, -2, -4 … Σ 1, -3, 5, -3, -11, 45, -91, 93, …
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A064861 → reverse → D+1
5, 5, 1 · 12, 18, 8, 1 · 16, 38, 31, 10, 1 … Σ 11, 39, 96, 324, 756, 2484, 5616, …
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A166962 → reverse
1 · 1, 1 · 1, 12, 1 · 1, 69, 103, 1 … Σ 1, 2, 14, 174, 3322, 89756, 3211420, …
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A008278 → x ↦ x+1 → D+1
1 · 3, 1 · 10, 7, 1 · 38, 43, 13, 1 … Σ 1, 4, 18, 95, 579, 3966, 29944, 246141, …
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A062264 → x ↦ x+1 → reverse
1 · 5, 6 · 15, 42, 28 · 35, 168, 252, 120 … Σ 1, 11, 85, 575, 3649, 22363, 134245, …
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A209414 → θ+1
1 · 1, 2 · 1, 8, 3 · 1, 14, 27, 4 … Σ 1, 3, 12, 46, 168, 594, 2052, 6966, …
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A102537 → (θ+1)²
1 · 1, 12 · 1, 32, 108 · 1, 60, 495, 880 … Σ 1, 13, 141, 1436, 14150, 136599, …
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A063967 → basis A013609 → θ+1
1 · 2, 4 · 6, 20, 12 · 16, 80, 96, 32 … Σ 1, 6, 38, 224, 1280, 7136, 39072, …
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A257614 → x ↦ x/(x+1)
1 · 2, 4 · 4, 36, 36 · 8, 268, 756, 504 … Σ 1, 6, 76, 1536, 42136, 1459176, …
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A038271 → basis A013610 → D+1
1 · 27, 15 · 504, 810, 225 · 8208, 22680, 18225, 3375 … Σ 1, 42, 1539, 52488, 1712421, 54206982, …
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A013610 → Veronese 2 (j=1) → D+1
3 · 6 · 36, 27 · 120, 108 … Σ 0, 3, 6, 63, 228, 1284, 5472, 25968, …
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A281000 → x ↦ x/(x+1)
1 · 3, 4 · 10, 30, 21 · 35, 175, 266, 127 … Σ 1, 7, 61, 603, 6457, 72975, 857013, …
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A120910 → D → D+1
3 · 18, 6 · 72, 54, 9 · 240, 288, 108, 12 … Σ 0, 3, 24, 135, 648, 2835, 11664, 45927, …
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basis-map coefficients of A021012 → basis A008277 → D
-1 · 6, 4 · -42, -72, -18 · 360, 1200, 720, 96 … Σ 0, -1, 10, -132, 2376, -56520, 1717920, …
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A063967 → basis A013609 → x ↦ x+1
1 · 4, 2 · 20, 18, 4 · 96, 128, 56, 8 … Σ 1, 6, 42, 288, 1980, 13608, 93528, …
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A156992 → basis A013610
1 · 4, 6 · 24, 72, 54 · 192, 864, 1296, 648 … Σ 1, 10, 150, 3000, 75000, 2250000, …
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A191935 → x ↦ x/(x+1)
1 · 1, 3 · 1, 10, 13 · 1, 23, 95, 81 … Σ 1, 4, 24, 200, 2160, 29088, 474688, …