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
- 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 2 2 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
- derivative_polynomial, shift -1 1 proof steps, 1 sequences
8,704 sequences
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A225117 → θ+1
1 · 2, 2 · 4, 26, 3 · 8, 186, 180, 4 … Σ 1, 4, 33, 378, 5508, 97200, 2012040, …
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A028326 → basis A013611 → (θ+1)²
2 · 4, 32 · 8, 128, 288 · 16, 384, 1728, 2048 … Σ 2, 36, 424, 4176, 37152, 309312, …
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A112486 → D
1 · 5, 6 · 26, 70, 45 · 154, 680, 945, 420 … Σ 0, 1, 11, 141, 2199, 40937, 889699, …
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A269944 → θ+1
1 · 0, 2 · 0, 2, 3 · 0, 8, 15, 4 … Σ 1, 2, 5, 27, 280, 4860, 128060, …
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A086645 → γ → D
4 · 12 · 24, 32 · 40, 160 … Σ 0, 0, 4, 12, 56, 200, 732, 2548, …
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A147716 → θ+1
1 · 14, 2 · 196, 56, 3 · 2744, 1176, 126, 4 … Σ 1, 16, 255, 4050, 64125, 1012500, …
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A136124 → x ↦ x+1 → D+1
1 · 4, 1 · 19, 9, 1 · 107, 71, 15, 1 … Σ 1, 5, 29, 194, 1478, 12664, 120696, …
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A074246 → D
2 · 12, 6 · 70, 60, 12 · 450, 510, 180, 20 … Σ 0, 2, 18, 142, 1160, 10208, 97720, …
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A028338 → x ↦ x+1
1 · 2, 1 · 8, 6, 1 · 48, 44, 12, 1 … Σ 1, 3, 15, 105, 945, 10395, 135135, …
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A285061 → reverse
1 · 4, 1 · 16, 24, 1 · 64, 240, 124, 1 … Σ 1, 5, 41, 429, 5329, 75989, 1215481, …
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basis-map coefficients of A112227 → basis A013609
1, 4, 4 · 1, 6, 12, 8 · 1, 8, 24, 32, 16 … Σ 9, 27, 81, 243, 729, 2187, 6561, 19683, …
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A069466: Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly tou…
1 · 2, 2 · 6, 24, 6 · 20, 180, 180, 20 … Σ 1, 4, 36, 400, 4900, 63504, 853776, …
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A104029 → basis A007318
1 · 3, 1 · 9, 7, 1 · 27, 34, 12, 1 … Σ 1, 4, 17, 74, 325, 1432, 6317, 27878, …
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basis-map coefficients of A112227 → basis A013611 → D
8, 32 · 12, 96, 192 · 16, 192, 768, 1024 … Σ 40, 300, 2000, 12500, 75000, 437500, …
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A257627 → D+1
1 · 6, 3 · 69, 78, 9 · 780, 2259, 834, 27 … Σ 1, 9, 156, 3900, 126360, 5012280, …
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A165253 → x ↦ x+1 → (θ+1)²
1 · 1 · 2, 4 · 5, 20, 9 … Σ 1, 1, 6, 34, 177, 873, 4146, 19142, …
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A094305 → x ↦ x+1 → D
3 · 24, 12 · 120, 120, 30 · 480, 720, 360, 60 … Σ 0, 3, 36, 270, 1620, 8505, 40824, …
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A013610 → basis A013610 → x ↦ x/(x+1)
1 · 4, 13 · 16, 104, 169 · 64, 624, 2028, 2197 … Σ 1, 17, 289, 4913, 83521, 1419857, …
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A391045 → reverse → D+1
1 · 2, 1 · 5, 6, 1 · 11, 28, 12, 1 … Σ 1, 3, 12, 52, 236, 1102, 5245, 25307, …
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A038303 → (θ+1)²
1 · 10, 4 · 100, 80, 9 · 1000, 1200, 270, 16 … Σ 1, 14, 189, 2486, 32065, 407286, …
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A099040 → basis A013610 → reverse
1 · 6, 2 · 36, 30, 6 · 216, 288, 120, 16 … Σ 1, 8, 72, 640, 5696, 50688, 451072, …
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A013609 → basis A013609 → θ+1
1 · 3, 8 · 9, 48, 48 · 27, 216, 432, 256 … Σ 1, 11, 105, 931, 7889, 64827, 521017, …
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A106246 → D
1 · 4, 2 · 3, 12, 3 · 0, 12, 24, 4 … Σ 0, 1, 6, 18, 40, 75, 126, 196, …
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A028326 → Veronese 2 (j=1) → D+1
2 · 4 · 8, 2 · 16, 8 … Σ 0, 2, 4, 10, 24, 56, 128, 288, …
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A078812 → basis A013610 → D
3 · 18, 18 · 75, 162, 81 · 270, 918, 972, 324 … Σ 0, 3, 36, 318, 2484, 18153, 127224, …
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basis-map coefficients of A127647 → D
1 · 4, 4 · 9, 18, 9 · 20, 60, 60, 20 … Σ 0, 1, 8, 36, 160, 640, 2496, 9408, …
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A143491 → basis A008277 → θ+1
12, 16, 3 · 60, 120, 45, 4 · 360, 960, 540, 96, 5 … Σ 31, 229, 1961, 19081, 207775, 2501801, …
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A347823 → θ+1
1 · 2, 6 · 3, 16, 15 · 4, 30, 54, 28 … Σ 1, 8, 34, 116, 352, 992, 2656, 6848, …
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A290319 → D+1
1 · 2, 1 · 11, 8, 1 · 104, 89, 18, 1 … Σ 1, 3, 20, 212, 3088, 57264, 1290048, …
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A113413 → basis A013610 → θ+1
1 · 3, 6 · 7, 36, 27 · 17, 138, 243, 108 … Σ 1, 9, 70, 506, 3494, 23394, 153182, …
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A166960 → x ↦ x+1
1 · 2, 1 · 8, 8, 1 · 50, 72, 24, 1 … Σ 1, 3, 17, 147, 1785, 28611, 580929, …
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A091044 → basis A013611 → (θ+1)²
1 · 4, 32 · 16, 256, 432 · 64, 1536, 5760, 4096 … Σ 1, 36, 704, 11456, 170752, 2408448, …
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A120910 → Veronese 2 → (θ+1)²
3 · 6 · 12, 12 · 24, 72 … Σ 3, 6, 24, 96, 363, 1326, 4740, 16680, …
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A038291 → basis A013610 → reverse
1 · 3, 10 · 9, 60, 100 · 27, 270, 900, 1000 … Σ 1, 13, 169, 2197, 28561, 371293, …
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A001263 → basis A013610 → reverse
1 · 3, 2 · 9, 15, 5 · 27, 81, 63, 14 … Σ 1, 5, 29, 185, 1257, 8925, 65445, …
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A013610 → D
3 · 6, 18 · 9, 54, 81 · 12, 108, 324, 324 … Σ 0, 3, 24, 144, 768, 3840, 18432, 86016, …
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A082985 → reverse
1 · 3, 1 · 5, 5, 1 · 7, 14, 7, 1 … Σ 1, 4, 11, 29, 76, 199, 521, 1364, …
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A038314 → x ↦ x+1 → basis A007318
1156, 816, 144 · 39304, 41616, 14688, 1728 … Σ 2116, 97336, 4477456, 205962976, …
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A038338 → basis A013609
576, 1152, 576 · 13824, 41472, 41472, 13824 … Σ 2304, 110592, 5308416, 254803968, …
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A085881 → basis A013611 → θ+1
1 · 2, 8 · 12, 96, 144 · 120, 1440, 4320, 3840 … Σ 1, 10, 252, 9720, 498960, 31842720, …
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A151775 → D+1
1 · 3, 2 · 33, 76, 24 · 723, 3302, 3480, 720 … Σ 1, 5, 133, 8225, 906313, 155522045, …
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A156992 → θ+1
1 · 2, 4 · 6, 24, 18 · 24, 144, 216, 96 … Σ 1, 6, 48, 480, 5760, 80640, 1290240, …
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A126351 → reverse → D
1 · 5, 2 · 19, 18, 3 · 65, 110, 42, 4 … Σ 0, 1, 7, 40, 221, 1241, 7218, 43807, …
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A128100 → basis A007318
1 · 1 · 3, 1 · 5, 2 … Σ 1, 1, 4, 7, 19, 40, 97, 217, …
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A008306 → θ+1
1 · 2 · 6, 6 · 24, 40 … Σ 1, 2, 12, 64, 425, 3198, 27216, 258144, …
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A159854 → x ↦ x+1 → D+1
3, 4, 1 · 7, 13, 7, 1 · 16, 38, 31, 10, 1 … Σ 8, 28, 96, 324, 1080, 3564, 11664, …
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A119462 → θ+1
1 · 2 · 2, 4 · 2, 12 … Σ 1, 2, 6, 14, 32, 72, 160, 352, …
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A178821 → basis A007318
1 · 10, 5 · 60, 60, 15 · 280, 420, 210, 35 … Σ 1, 15, 135, 945, 5670, 30618, 153090, …
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A106800 → (θ+1)²
1 · 1 · 1, 4 · 1, 12, 9 … Σ 1, 1, 5, 22, 104, 531, 2897, 16787, …
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A272866 → (θ+1)²
1 · 1, 12, 9 · 1, 24, 99, 96, 25 … Σ 1, 22, 245, 2150, 16625, 118750, …