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_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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A114687 → basis A013609 → x ↦ x+1
1 · 7, 4 · 55, 60, 16 · 469, 744, 384, 64 … Σ 1, 11, 131, 1661, 22101, 305151, …
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A008288 → basis A013611 → θ+1
1 · 2, 8 · 5, 40, 48 · 12, 144, 384, 256 … Σ 1, 10, 93, 796, 6509, 51582, 399665, …
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type_b_eulerian → γ → (θ+1)²
1 · 1 · 1, 16 · 1, 80 … Σ 1, 1, 17, 81, 1009, 9713, 134305, …
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A111125: Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
1 · 3, 1 · 5, 5, 1 · 7, 14, 7, 1 … Σ 1, 4, 11, 29, 76, 199, 521, 1364, …
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A038207 → basis A008277 → (θ+1)²
1 · 3, 4 · 9, 28, 9 · 27, 148, 108, 16 … Σ 1, 7, 46, 299, 1967, 13248, 91903, …
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delannoy → basis A013609 → D+1
1 · 4, 2 · 15, 18, 4 · 48, 100, 56, 8 … Σ 1, 6, 37, 212, 1173, 6322, 33425, …
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A111125 → D → D+1
1 · 7, 2 · 28, 20, 3 · 84, 108, 39, 4 … Σ 0, 1, 9, 51, 235, 960, 3624, 12941, …
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reversed rows → D+1
1 · 3, 2 · 4, 3 · 5, 8, 2 … Σ 1, 5, 7, 15, 26, 48, 85, 151, …
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A352880 → reverse → θ+1
8 · 18, 24 · 32, 128, 48 · 50, 400, 450, 80 … Σ 8, 42, 208, 980, 4464, 19866, 86944, …
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A038319 → basis A013609 → (θ+1)²
256, 1280, 900 · 4096, 30720, 43200, 16000 … Σ 2436, 94016, 3377296, 115579776, …
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A114656 → reverse → x ↦ x/(x+1)
1 · 1, 3 · 1, 8, 11 · 1, 15, 51, 45 … Σ 1, 4, 20, 112, 672, 4224, 27456, …
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A114656 → reverse → x ↦ x+1
1 · 3, 2 · 11, 14, 4 · 45, 84, 48, 8 … Σ 1, 5, 29, 185, 1257, 8925, 65445, …
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A249250 → D
3 · 6, 6 · 9, 30, 9 · 12, 72, 72, 12 … Σ 0, 3, 12, 48, 168, 564, 1824, 5760, …
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A091042 → basis A013610 → D+1
1 · 13, 9 · 76, 150, 45 · 400, 1344, 1071, 189 … Σ 1, 22, 271, 3004, 31981, 332146, …
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A013612 → basis A013610
36, 180, 225 · 216, 1620, 4050, 3375 … Σ 441, 9261, 194481, 4084101, 85766121, …
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A035342 → reverse
1 · 1, 3 · 1, 9, 15 · 1, 18, 87, 105 … Σ 1, 4, 25, 211, 2236, 28471, 422899, …
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A249251 → basis A007318 → D
3 · 18, 12 · 87, 114, 36 · 378, 738, 468, 96 … Σ 0, 3, 30, 237, 1680, 11211, 71970, …
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A038303 → basis A007318
1 · 11, 1 · 121, 22, 1 · 1331, 363, 33, 1 … Σ 1, 12, 144, 1728, 20736, 248832, …
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basis-map coefficients of A297191, shifted → basis A013610 → x ↦ x+1
1 · 16, 24, 9 · 256, 768, 864, 432, 81 … Σ 1, 49, 2401, 117649, 5764801, …
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A038324 → basis A013609 → reverse
1 · 20, 21 · 400, 840, 441 · 8000, 25200, 26460, 9261 … Σ 1, 41, 1681, 68921, 2825761, 115856201, …
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A130534 → basis A008277 → (θ+1)²
1 · 2, 4 · 6, 24, 9 · 24, 144, 108, 16 … Σ 1, 6, 39, 292, 2505, 24306, 263431, …
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narayana → basis A013610 → θ+1
1 · 2, 6 · 5, 30, 27 · 14, 126, 243, 108 … Σ 1, 8, 62, 491, 3975, 32730, 272940, …
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A204420 → basis A013610 → θ+1
1 · 1, 6 · 9, 72, 81 · 225, 2070, 3645, 1620 … Σ 1, 7, 162, 7560, 589680, 69128640, …
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A139548 → basis A013609
1 · 8, 16, 8 · 64, 256, 384, 256, 64 … Σ 1, 32, 1024, 32768, 1048576, 33554432, …
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A111125 → basis A007318 → (θ+1)²
1 · 4, 4 · 11, 28, 9 · 29, 124, 90, 16 … Σ 1, 8, 48, 259, 1309, 6328, 29616, …
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A207606 → x ↦ x+1 → θ+1
13, 22, 6 · 34, 88, 51, 8 · 89, 310, 279, 92, 10 … Σ 41, 181, 780, 3301, 13775, 56841, …
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A008288 → D
1 · 3, 2 · 5, 10, 3 · 7, 26, 21, 4 … Σ 0, 1, 5, 18, 58, 175, 507, 1428, …
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A060920 → Veronese 2 (j=1) → D
1 · 14 · 105, 2 · 594, 54 … Σ 0, 0, 0, 1, 14, 107, 648, 3493, …
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A038305 → basis A013609 → θ+1
169, 312, 108 · 2197, 6084, 4212, 864 … Σ 589, 13357, 294937, 6385729, 136185445, …
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A081579 → x ↦ x+1 → D
1 · 8, 2 · 36, 28, 3 · 160, 200, 60, 4 … Σ 0, 1, 10, 67, 424, 2469, 13926, 76199, …
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A272867 → basis A013610 → reverse
1 · 9, 18, 6 · 81, 324, 432, 216, 36 … Σ 1, 33, 1089, 35937, 1185921, 39135393, …
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A038327 → basis A013611 → (θ+1)²
1 · 13, 16 · 169, 416, 144 · 2197, 8112, 5616, 1024 … Σ 1, 29, 729, 16949, 374833, 8003277, …
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A094816: Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.
1 · 1, 1 · 1, 3, 1 · 1, 8, 6, 1 … Σ 1, 2, 5, 16, 65, 326, 1957, 13700, …
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A347056: Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.
1 · 3, 4 · 6, 16, 10 · 10, 40, 50, 20 … Σ 1, 7, 32, 120, 400, 1232, 3584, 9984, …
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A142461 → D+1
1 · 2, 1 · 15, 16, 1 · 112, 333, 114, 1 … Σ 1, 3, 32, 560, 13440, 407680, 14909440, …
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A257609 → D (shift +2) → x ↦ x/(x+1)
11, 44, 36 · 26, 210, 420, 240 … Σ 91, 896, 9829, 120198, 1628911, …
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A344565 → x ↦ x/(x+1)
3 · 15, 30 · 45, 180, 180 · 105, 630, 1260, 840 … Σ 3, 45, 405, 2835, 17010, 91854, 459270, …
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A078812 → x ↦ x+1 → (θ+1)²
1 · 3, 4 · 8, 24, 9 · 21, 100, 81, 16 … Σ 1, 7, 41, 218, 1091, 5237, 24379, …
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A125092 → θ+1 → x ↦ x/(x+1)
1 · 1, 9 · 1, 18, 44 · 1, 27, 132, 170 … Σ 1, 10, 63, 330, 1557, 6858, 28755, …
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A193685 → D
1 · 11, 2 · 91, 36, 3 · 671, 434, 78, 4 … Σ 0, 1, 13, 130, 1187, 10451, 90954, …
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A038335 → basis A013609 → reverse
1 · 18, 21 · 324, 756, 441 · 5832, 20412, 23814, 9261 … Σ 1, 39, 1521, 59319, 2313441, 90224199, …
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A132440 → θ+1 → x ↦ x+1
1 · 4, 4 · 9, 18, 9 · 16, 48, 48, 16 … Σ 0, 1, 8, 36, 128, 400, 1152, 3136, …
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A035607 → basis A013609 → θ+1
1 · 1 · 3, 8 · 7, 32, 24 … Σ 1, 1, 11, 63, 361, 1985, 10659, 56191, …
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A225470 → basis A013609 → (θ+1)²
1 · 3, 8 · 18, 72, 36 · 162, 792, 648, 128 … Σ 1, 11, 126, 1730, 28324, 542484, …
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A086915 → D+1
2 · 8, 4 · 36, 40, 8 · 192, 336, 144, 16 … Σ 2, 12, 84, 688, 6480, 69024, 820064, …
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A038303 → basis A007318 → (θ+1)²
1 · 11, 4 · 121, 88, 9 · 1331, 1452, 297, 16 … Σ 1, 15, 218, 3096, 43200, 594432, …
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A038321 → basis A007318
324, 252, 49 · 5832, 6804, 2646, 343 … Σ 625, 15625, 390625, 9765625, 244140625, …
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A225470 → basis A013609 → reverse
1 · 2, 3 · 4, 18, 18 · 8, 72, 198, 162 … Σ 1, 5, 40, 440, 6160, 104720, 2094400, …
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A028326 → basis A013610 → D+1
2 · 10, 6 · 32, 60, 18 · 88, 288, 270, 54 … Σ 2, 16, 110, 700, 4250, 25000, 143750, …
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A127648 → Veronese 2 → x ↦ x+1
1 · 3, 3 · 5, 10, 5 · 7, 21, 21, 7 … Σ 0, 1, 0, 6, 0, 20, 0, 56, …