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
- 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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A330510 → θ+1
2 · 3, 6 · 4, 20, 12 · 5, 44, 66, 20 … Σ 2, 9, 36, 135, 486, 1701, 5832, 19683, …
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A121574 → reverse → θ+1
1 · 1, 4 · 1, 10, 12 · 1, 16, 48, 32 … Σ 1, 5, 23, 97, 390, 1518, 5773, 21575, …
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A161198 → basis A008277 → (θ+1)²
1 · 3, 8 · 15, 80, 36 · 105, 840, 756, 128 … Σ 1, 11, 131, 1829, 29641, 548347, …
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A178822 → D+1
1 · 12, 6 · 63, 84, 21 · 224, 504, 336, 56 … Σ 1, 18, 168, 1120, 6048, 28224, 118272, …
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A166340 → (θ+1)²
1, 32, 9 · 1, 76, 171, 16 · 1, 168, 1026, 672, 25 … Σ 42, 264, 1892, 15368, 139840, 1410240, …
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A038324 → basis A007318 → D
10 · 420, 200 · 13230, 12600, 3000 … Σ 0, 10, 620, 28830, 1191640, 46176050, …
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basis-map coefficients of A026729 → basis A013609 → x ↦ x+1
1 · 3, 2 · 9, 12, 4 · 27, 54, 36, 8 … Σ 1, 5, 25, 125, 625, 3125, 15625, 78125, …
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A124848 → basis A007318
1 · 5, 4 · 19, 28, 10 · 63, 132, 90, 20 … Σ 1, 9, 57, 305, 1473, 6633, 28377, …
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A201637 → reverse (shift +1) → (θ+1)²
1 · 0, 4 · 0, 8, 9 · 0, 24, 72, 16 … Σ 1, 4, 17, 112, 995, 11060, 147105, …
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A013622 → basis A007318 → x ↦ x/(x+1)
1 · 8, 13 · 64, 208, 169 · 512, 2496, 4056, 2197 … Σ 1, 21, 441, 9261, 194481, 4084101, …
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basis-map coefficients of A112227 → x ↦ x+1 → basis A008277
4, 5, 1 · 8, 19, 9, 1 · 16, 65, 55, 14, 1 … Σ 10, 37, 151, 674, 3263, 17007, 94828, …
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A038314 → basis A013609 → D+1
1540, 2208, 576 · 45496, 110880, 79488, 13824 … Σ 4324, 249688, 13821712, 743257696, …
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A122542 → D+1
1 · 1, 1 · 2, 4, 1 · 2, 10, 7, 1 … Σ 1, 2, 7, 20, 57, 158, 431, 1160, …
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A039756 → D+1
1 · 2, 1 · 5, 6, 1 · 10, 35, 16, 1 … Σ 1, 3, 12, 62, 372, 2492, 18384, 147800, …
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A103327 → basis A013610 → θ+1
1 · 4, 6 · 16, 72, 27 · 64, 480, 648, 108 … Σ 1, 10, 115, 1300, 14149, 149470, …
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A028338 → basis A013611 → (θ+1)²
1 · 2, 16 · 8, 96, 144 · 48, 704, 1728, 1024 … Σ 1, 18, 248, 3504, 53824, 910208, …
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A109692 → basis A013609 → θ+1
1 · 2, 4 · 8, 40, 36 · 48, 400, 816, 480 … Σ 1, 6, 84, 1744, 47328, 1578624, …
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A197208 → x ↦ x/(x+1)
30, 180, 180 · 60, 780, 1980, 1320 … Σ 390, 4140, 43785, 464184, 4941468, …
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A026374 → D+1 → D+1
1 · 3, 1 · 9, 7, 1 · 17, 26, 10, 1 … Σ 1, 4, 17, 54, 195, 540, 1775, 4550, …
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A137320 → Veronese 2 → D+1
1 · 4, 4 · 12, 12 · 44, 76, 16 … Σ 1, 0, 8, 24, 136, 880, 6528, 54768, …
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A027465 → basis A008277 → reverse
1 · 1, 4 · 1, 9, 16 · 1, 15, 61, 64 … Σ 1, 5, 26, 141, 799, 4736, 29371, …
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A114687 → basis A007318 → x ↦ x+1
1 · 5, 2 · 29, 22, 4 · 185, 204, 72, 8 … Σ 1, 7, 55, 469, 4237, 39907, 387739, …
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A046739 → D+1
8, 9, 1 · 22, 63, 24, 1 · 52, 373, 314, 55, 1 … Σ 18, 110, 795, 6489, 59332, 600732, …
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A102537 → x ↦ x/(x+1)
1 · 1, 4 · 1, 10, 21 · 1, 18, 88, 126 … Σ 1, 5, 32, 233, 1833, 15180, 130392, …
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A102662 → x ↦ x+1
1 · 4, 3 · 9, 11, 3 · 22, 38, 20, 3 … Σ 1, 7, 23, 83, 295, 1051, 3743, 13331, …
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A008459 → D → (θ+1)²
1 · 4, 8 · 9, 72, 27 · 16, 288, 432, 64 … Σ 0, 1, 12, 108, 800, 5250, 31752, …
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A269951 → D+1
1 · 1, 1 · 2, 4, 1 · 5, 15, 8, 1 … Σ 1, 2, 7, 29, 142, 814, 5391, 40689, …
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A028421 → D+1
1 · 3, 2 · 8, 12, 3 · 28, 58, 30, 4 … Σ 1, 5, 23, 120, 724, 5012, 39332, …
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A086645 → ⊙ C(n+1,k) (shift +2) → x ↦ x+1
14, 14, 1 · 92, 138, 48, 1 · 646, 1292, 762, 116, 1 … Σ 29, 279, 2817, 29283, 309869, 3321447, …
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A159854 → basis A013609 → θ+1
1, 8, 12 · 2, 20, 48, 32 · 4, 48, 156, 192, 80 … Σ 21, 102, 480, 2208, 9984, 44544, …
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A137597 → x ↦ x+1
1 · 3, 1 · 11, 7, 1 · 47, 43, 12, 1 … Σ 1, 4, 19, 103, 622, 4117, 29521, …
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basis-map coefficients of A297191, shifted → basis A013609 → D
4, 8 · 8, 48, 96, 64 · 12, 120, 480, 960, 960, 384 … Σ 0, 12, 216, 2916, 34992, 393660, …
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A112007 → D+1
1 · 3, 1 · 14, 10, 1 · 82, 102, 25, 1 … Σ 1, 4, 25, 210, 2205, 27720, 405405, …
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A038329 → basis A013610 → reverse
81, 270, 225 · 729, 3645, 6075, 3375 … Σ 576, 13824, 331776, 7962624, 191102976, …
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A013620 → (θ+1)²
1 · 2, 12 · 4, 48, 81 · 8, 144, 486, 432 … Σ 1, 14, 133, 1070, 7825, 53750, 353125, …
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A200536 → basis A013609 → x ↦ x/(x+1)
1 · 6, 26, 28 · 36, 312, 1012, 1456, 784 … Σ 1, 60, 3600, 216000, 12960000, …
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A321434 → θ+1
1 · 0, 2 · 0, 2 · 0, 2, 6 … Σ 1, 2, 2, 8, 8, 44, 44, 308, …
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A225471 → basis A007318
1 · 4, 1 · 32, 12, 1 · 384, 176, 24, 1 … Σ 1, 5, 45, 585, 9945, 208845, 5221125, …
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A391403 → D
2 · 6 · 12, 12 · 20, 60 … Σ 0, 0, 2, 6, 24, 80, 270, 882, …
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A257626: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.
1 · 6, 6 · 36, 108, 36 · 216, 1404, 1404, 216 … Σ 1, 12, 180, 3240, 68040, 1632960, …
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A134082 → reverse
1 · 1, 2 · 1, 4 · 1, 6 … Σ 1, 3, 5, 7, 9, 11, 13, 15, …
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A027466 → basis A013610 → x ↦ x/(x+1)
1 · 8, 11 · 64, 176, 121 · 512, 2112, 2904, 1331 … Σ 1, 19, 361, 6859, 130321, 2476099, …
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A008306 → D+1
1 · 2 · 9, 3 · 44, 20 … Σ 1, 2, 12, 64, 425, 3198, 27216, 258144, …
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A116666: Triangle, row sums = number of edges in n-dimensional hypercubes.
1 · 1, 3 · 1, 6, 5 · 1, 9, 15, 7 … Σ 1, 4, 12, 32, 80, 192, 448, 1024, …
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A122542 → (θ+1)²
1 · 0, 4 · 0, 8, 9 · 0, 8, 36, 16 … Σ 1, 4, 17, 60, 201, 640, 1969, 5896, …
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A116412 → basis A007318 → D+1
1 · 5, 1 · 21, 10, 1 · 85, 66, 15, 1 … Σ 1, 6, 32, 167, 852, 4276, 21184, …
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A001263 → basis A013610 → D
3 · 15, 18 · 63, 162, 81 · 252, 1008, 1134, 324 … Σ 0, 3, 33, 306, 2718, 23805, 207495, …
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A026729 → basis A013609 → x ↦ x/(x+1)
1 · 1, 3 · 2, 10, 12 · 3, 23, 57, 45 … Σ 1, 4, 24, 128, 704, 3840, 20992, …
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A052553 → x ↦ x+1
1 · 1 · 2, 1 · 3, 2 … Σ 1, 1, 3, 5, 11, 21, 43, 85, …
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A088996 → basis A007318 → (θ+1)²
1 · 1, 4 · 3, 20, 18 · 15, 136, 225, 96 … Σ 1, 5, 41, 472, 6968, 125172, 2644740, …