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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A120910 → basis A007318
3 · 9, 3 · 27, 18, 3 · 81, 81, 27, 3 … Σ 3, 12, 48, 192, 768, 3072, 12288, …
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A360282 → x ↦ x/(x+1)
1 · 0, 1 · 0, 3, 5 · 0, 10, 32, 25 … Σ 1, 1, 8, 67, 584, 5221, 47480, 437095, …
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A062145: Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.
1 · 1, 4 · 1, 10, 10 · 1, 18, 45, 20 … Σ 1, 5, 21, 84, 330, 1287, 5005, 19448, …
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image under θ+1 → θ+1 → D
4 · 0, 18 · 0, 0, 48 · 0, 0, 0, 100 … Σ 0, 4, 18, 48, 100, 180, 294, 448, …
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A161198 → basis A013610 → D
6 · 48, 72 · 426, 1080, 648 … Σ 0, 6, 120, 2154, 40032, 796374, …
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A001263 → θ+1 → D
2 · 6, 6 · 12, 36, 12 · 20, 120, 120, 20 … Σ 0, 2, 12, 60, 280, 1260, 5544, 24024, …
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A135065 → basis A013609
1 · 8, 8 · 36, 72, 36 · 128, 384, 384, 128 … Σ 1, 16, 144, 1024, 6400, 36864, 200704, …
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A056241 → x ↦ x/(x+1)
1 · 1, 2 · 1, 5, 5 · 1, 9, 21, 14 … Σ 1, 3, 11, 45, 193, 843, 3707, 16341, …
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A111006 → x ↦ x/(x+1)
1 · 0, 1 · 0, 1, 3 · 0, 0, 2, 5 … Σ 1, 1, 4, 7, 19, 40, 97, 217, …
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A147716 → basis A013609 → D
2 · 60, 8 · 1350, 360, 24 · 27000, 10800, 1440, 64 … Σ 0, 2, 68, 1734, 39304, 835210, …
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A046716 → D+1
1 · 2, 1 · 4, 5, 1 · 7, 22, 11, 1 … Σ 1, 3, 10, 41, 209, 1287, 9270, 76225, …
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A225471 → basis A013610 → θ+1
1 · 4, 6 · 32, 72, 27 · 384, 1056, 648, 108 … Σ 1, 10, 131, 2196, 45189, 1105182, …
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A028326 → basis A013610 → (θ+1)²
2 · 4, 24 · 8, 96, 162 · 16, 288, 972, 864 … Σ 2, 28, 266, 2140, 15650, 107500, …
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A038271 → basis A007318 → D
5 · 120, 50 · 2160, 1800, 375 … Σ 0, 5, 170, 4335, 98260, 2088025, …
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A013611 → basis A013611 → x ↦ x/(x+1)
1 · 5, 21 · 25, 210, 441 · 125, 1575, 6615, 9261 … Σ 1, 26, 676, 17576, 456976, 11881376, …
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A156365 → reverse
4, 8, 1 · 8, 44, 22, 1 · 16, 208, 264, 52, 1 … Σ 13, 75, 541, 4683, 47293, 545835, …
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A143495: Triangle read by rows: 3-Stirling numbers of the second kind.
1 · 3, 1 · 9, 7, 1 · 27, 37, 12, 1 … Σ 1, 4, 17, 77, 372, 1915, 10481, 60814, …
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A121574 → θ+1
1 · 2, 2 · 4, 10, 3 · 8, 32, 24, 4 … Σ 1, 4, 17, 68, 264, 1002, 3739, 13768, …
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A165253 → x ↦ x+1 → D+1
1 · 1 · 3, 1 · 10, 7, 1 … Σ 1, 1, 4, 18, 79, 339, 1430, 5952, …
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A196347 → D
1 · 4, 4 · 18, 36, 18 · 96, 288, 288, 96 … Σ 0, 1, 8, 72, 768, 9600, 138240, …
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A137320 → reverse → D
2 · 12, 8 · 48, 88, 36 · 160, 560, 600, 192 … Σ 0, 0, 2, 20, 172, 1512, 14184, 143712, …
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A028326 → basis A013610 → x ↦ x+1
2 · 10, 6 · 50, 60, 18 · 250, 450, 270, 54 … Σ 2, 16, 128, 1024, 8192, 65536, 524288, …
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chebyshev_u → x ↦ x+1
1 · 2, 2 · 3, 8, 4 · 4, 20, 24, 8 … Σ 1, 4, 15, 56, 209, 780, 2911, 10864, …
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A136124 → basis A013609 → (θ+1)²
1 · 3, 8 · 12, 56, 36 · 60, 376, 432, 128 … Σ 1, 11, 104, 996, 10084, 109364, …
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A172431 → D → θ+1
2 · 4, 12 · 6, 40, 36 · 8, 84, 180, 80 … Σ 0, 2, 16, 82, 352, 1368, 4980, 17302, …
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A030528 → basis A013609 → θ+1
1 · 2, 4 · 3, 16, 12 · 5, 40, 72, 32 … Σ 1, 6, 31, 149, 684, 3045, 13257, 56754, …
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A225471 → basis A008277 → (θ+1)²
1 · 4, 4 · 32, 52, 9 · 384, 804, 243, 16 … Σ 1, 8, 93, 1447, 28292, 666681, …
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A013612 → (θ+1)²
1, 40, 225 · 1, 60, 675, 2000 … Σ 266, 2736, 25056, 212976, 1718496, …
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A013610 → basis A013610
1 · 4, 9 · 16, 72, 81 · 64, 432, 972, 729 … Σ 1, 13, 169, 2197, 28561, 371293, …
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A099097 → reverse → D+1
1 · 3 · 10, 1 · 33, 6 … Σ 1, 3, 11, 39, 138, 486, 1705, 5961, …
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A113413 → Veronese 2 (j=1) → x ↦ x+1
1 · 4 · 9, 1 · 20, 8 … Σ 0, 1, 4, 10, 28, 84, 244, 704, …
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A038319 → basis A013611 → x ↦ x/(x+1)
256, 1152, 1296 · 4096, 27648, 62208, 46656 … Σ 2704, 140608, 7311616, 380204032, …
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A154228: Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*T(n-2, k-1), read by rows.
1 · 1, 1 · 1, 16, 1 · 1, 47, 47, 1 … Σ 1, 2, 18, 96, 1182, 11100, 187680, …
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A111125 → reverse → D+1
1 · 4, 3 · 6, 15, 5 · 8, 35, 35, 7 … Σ 1, 7, 26, 85, 265, 804, 2393, 7019, …
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A225470 → basis A013609
1 · 3, 2 · 18, 18, 4 · 162, 198, 72, 8 … Σ 1, 5, 40, 440, 6160, 104720, 2094400, …
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A143685 → D+1
1 · 2, 1 · 12, 13, 1 · 22, 63, 24, 1 … Σ 1, 3, 26, 110, 615, 2821, 13828, 63756, …
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A106246 → reverse → D+1
1 · 3, 2 · 5, 6, 1 · 7, 12, 3 … Σ 1, 5, 12, 22, 35, 51, 70, 92, …
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A130534 → D
1 · 3, 2 · 11, 12, 3 · 50, 70, 30, 4 … Σ 0, 1, 5, 26, 154, 1044, 8028, 69264, …
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A133080 → basis A013609 → reverse
1 · 2, 2 · 4, 4, 1 · 8, 16, 10, 2 … Σ 1, 4, 9, 36, 81, 324, 729, 2916, …
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A200536 → D
3, 4 · 6, 26, 36, 16 · 9, 66, 189, 264, 180, 48 … Σ 0, 7, 84, 756, 6048, 45360, 326592, …
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A091044 → basis A013610 → (θ+1)²
1 · 4, 24 · 16, 192, 243 · 64, 1152, 3240, 1728 … Σ 1, 28, 451, 6184, 78301, 941428, …
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A007318 → −D²+(−1+n)D+1+n → D
2 · 8, 6 · 18, 36, 12 · 32, 108, 96, 20 … Σ 0, 2, 14, 66, 256, 880, 2784, 8288, …
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A028338 → basis A013611 → x ↦ x/(x+1)
1 · 2, 6 · 8, 40, 48 · 48, 320, 688, 480 … Σ 1, 8, 96, 1536, 30720, 737280, …
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A144443 → x ↦ x/(x+1)
1 · 1, 2 · 1, 22, 22 · 1, 162, 480, 320 … Σ 1, 3, 45, 963, 28413, 1078515, …
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A038314 → basis A013609 → x ↦ x/(x+1)
484, 2024, 2116 · 10648, 66792, 139656, 97336 … Σ 4624, 314432, 21381376, 1453933568, …
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A207607 → basis A013610 → reverse
1 · 6, 3 · 18, 27, 8 · 54, 135, 99, 21 … Σ 1, 9, 53, 309, 1801, 10497, 61181, …
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A008459 → basis A013610 → D
3 · 18, 18 · 90, 216, 81 · 420, 1620, 1620, 324 … Σ 0, 3, 36, 387, 3984, 39915, 392580, …
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A225471 → (θ+1)²
1 · 3, 4 · 21, 40, 9 · 231, 524, 189, 16 … Σ 1, 7, 70, 960, 16864, 362112, 9199616, …
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A166979 → x ↦ x+1
13, 13, 1 · 143, 174, 33, 1 · 1852, 2651, 861, 62, 1 … Σ 27, 351, 5427, 98415, 2060883, …
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A038321 → basis A007318 → x ↦ x/(x+1)
324, 900, 625 · 5832, 24300, 33750, 15625 … Σ 1849, 79507, 3418801, 147008443, …