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
- affine, shift -1 6 proof steps, 5 sequences
- hadamard_multiplier, shift 0 6 proof steps, 5 sequences
- affine, shift 1 5 proof steps, 4 sequences
- hadamard_multiplier, shift 2 4 proof steps, 3 sequences
- affine, shift 2 3 proof steps, 3 sequences
- euler_polynomial, shift 1 3 proof steps, 2 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
- 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
- hadamard_multiplier, shift 1 1 proof steps, 1 sequences
8,704 sequences
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A103633 → (θ+1)²
1 · 0, 4 · 0, 4, 9 · 0, 0, 9, 16 … Σ 1, 4, 13, 25, 66, 102, 248, 344, …
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A013610 → basis A007318 → D
3 · 24, 18 · 144, 216, 81 · 768, 1728, 1296, 324 … Σ 0, 3, 42, 441, 4116, 36015, 302526, …
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A078812 → x ↦ x+1 → 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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A257612: 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) = 4*x + 2.
1 · 2, 2 · 4, 24, 4 · 8, 184, 184, 8 … Σ 1, 4, 32, 384, 6144, 122880, 2949120, …
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A166979 → D+1
12, 13, 1 · 112, 171, 33, 1 · 1112, 2473, 855, 62, 1 … Σ 26, 317, 4503, 74074, 1400719, …
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A249250 → x ↦ x+1 → reverse
1 · 3, 4 · 3, 12, 10 · 3, 24, 48, 28 … Σ 1, 7, 25, 103, 409, 1639, 6553, 26215, …
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A039755 → (θ+1)²
1 · 1, 4 · 1, 16, 9 · 1, 52, 81, 16 … Σ 1, 5, 26, 150, 964, 6836, 52936, …
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A038320 → x ↦ x+1 → (θ+1)²
289, 816, 324 · 4913, 20808, 16524, 3456 … Σ 1429, 45701, 1384393, 40382273, …
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A232535 → basis A007318 → θ+1
1 · 3, 4 · 12, 28, 9 · 48, 160, 111, 16 … Σ 1, 7, 49, 335, 2241, 14743, 95729, …
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A163932: Triangle related to the asymptotic expansion of E(x,m=3,n).
A130534 → D²∘x (shift +1)1 · 3, 3 · 11, 18, 6 · 50, 105, 60, 10 … Σ 1, 6, 35, 225, 1624, 13132, 118124, …
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A039755 → x ↦ x+1
1 · 2, 1 · 6, 6, 1 · 24, 34, 12, 1 … Σ 1, 3, 13, 71, 457, 3355, 27509, 248127, …
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A116412 → D
1 · 6, 2 · 21, 18, 3 · 60, 90, 36, 4 … Σ 0, 1, 8, 42, 190, 797, 3192, 12391, …
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A135838 → (θ+1)²
1 · 2, 8 · 2, 16, 18 · 4, 48, 108, 64 … Σ 1, 10, 36, 224, 640, 3456, 8960, 45056, …
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A166972 → D+1
1 · 2, 1 · 11, 12, 1 · 84, 165, 44, 1 … Σ 1, 3, 24, 294, 5160, 123036, 3759876, …
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A120910 → Veronese 2 → x ↦ x+1
3 · 6 · 15, 3 · 42, 18 … Σ 3, 6, 18, 60, 204, 696, 2376, 8112, …
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A209414 → D+1
1 · 2, 1 · 5, 6, 1 · 8, 25, 12, 1 … Σ 1, 3, 12, 46, 168, 594, 2052, 6966, …
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A132812 → basis A007318 → basis A007318
1 · 6, 2 · 33, 21, 3 · 180, 168, 48, 4 … Σ 1, 8, 57, 400, 2810, 19824, 140497, …
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A216973 → θ+1 → D+1
1 · 6, 4 · 15, 30, 9 · 28, 96, 84, 16 … Σ 0, 1, 10, 54, 224, 800, 2592, 7840, …
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A123519 → basis A013611 → reverse
64, 56, 11 · 512, 704, 304, 41 … Σ 131, 1561, 18601, 221651, 2641211, …
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A038207 → basis A008277 → θ+1
1 · 3, 2 · 9, 14, 3 · 27, 74, 36, 4 … Σ 1, 5, 26, 141, 799, 4736, 29371, …
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A088996 → basis A013610 → x ↦ x/(x+1)
1 · 1, 4 · 3, 21, 36 · 15, 147, 474, 504 … Σ 1, 5, 60, 1140, 29640, 978120, …
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A347056 → reverse
1 · 4, 3 · 10, 16, 6 · 20, 50, 40, 10 … Σ 1, 7, 32, 120, 400, 1232, 3584, 9984, …
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Veronese parent of A109446 → Veronese 2 (j=1) → θ+1
1 · 1 · 1, 2 · 3, 2 … Σ 1, 1, 3, 5, 16, 28, 80, 144, …
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image under θ+1 → θ+1 → D+1 → D
4 · 18, 18 · 0, 96, 48 · 0, 0, 300, 100 … Σ 0, 4, 36, 144, 400, 900, 1764, 3136, …
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A075499 → (θ+1)²
16, 48, 9 · 64, 448, 216, 16 … Σ 73, 744, 8361, 102784, 1373121, …
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A372868 → D+1
1 · 2 · 5, 1 · 14, 6 … Σ 1, 2, 6, 20, 67, 222, 728, 2368, …
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A272866 → basis A007318
1 · 5, 5, 1 · 25, 50, 35, 10, 1 … Σ 1, 11, 121, 1331, 14641, 161051, …
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A126216 → basis A013610 → D+1
1 · 6, 3 · 32, 39, 9 · 171, 342, 189, 27 … Σ 1, 9, 80, 729, 6806, 64746, 624694, …
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basis-map coefficients of A021012 → (θ+1)²
1 · 0, -4 · 0, 0, 18 · 0, 0, 0, -96 … Σ 1, -4, 18, -96, 600, -4320, 35280, …
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A106246 → θ+1 → x ↦ x/(x+1)
1 · 2, 4 · 1, 10, 12 · 0, 6, 30, 28 … Σ 1, 6, 23, 64, 141, 266, 451, 708, …
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A038322 → basis A007318 → D+1
1 · 27, 8 · 665, 432, 64 · 15523, 15960, 5184, 512 … Σ 1, 35, 1161, 37179, 1161297, 35606547, …
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A144445: Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
1 · 1, 1 · 1, 23, 1 · 1, 206, 206, 1 … Σ 1, 2, 25, 414, 9697, 293808, 10938775, …
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A091044 → basis A013609 → x ↦ x/(x+1)
1 · 4, 8 · 16, 64, 60 · 64, 384, 736, 448 … Σ 1, 12, 140, 1632, 19024, 221760, …
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A007318 → −D²+(−1+n)D+1+n → x ↦ x/(x+1)
1 · 2, 4 · 3, 14, 14 · 4, 30, 66, 44 … Σ 1, 6, 31, 144, 621, 2538, 9963, 37908, …
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A056242 → θ+1
1 · 1, 4 · 1, 10, 12 · 1, 18, 48, 32 … Σ 1, 5, 23, 99, 408, 1632, 6388, 24596, …
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A102413 → D+1
1 · 2, 1 · 5, 6, 1 · 7, 18, 9, 1 … Σ 1, 3, 12, 35, 102, 287, 792, 2151, …
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A034870 → basis A013610 → x ↦ x/(x+1)
1 · 4, 20, 25 · 16, 160, 600, 1000, 625 … Σ 1, 49, 2401, 117649, 5764801, …
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image under θ+1 → θ+1 → (θ+1)²
1 · 0, 16 · 0, 0, 81 · 0, 0, 0, 256 … Σ 1, 16, 81, 256, 625, 1296, 2401, 4096, …
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A127773 → θ+1 → reverse
1 · 6 · 18 · 40 … Σ 1, 6, 18, 40, 75, 126, 196, 288, …
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A008288 → basis A013609 → θ+1
1 · 2, 4 · 5, 20, 12 · 12, 72, 96, 32 … Σ 1, 6, 37, 212, 1173, 6322, 33425, …
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Hadamard-factor parent of A155495 → x ↦ x+1
8, 8, 1 · 32, 48, 18, 1 · 128, 256, 160, 32, 1 … Σ 17, 99, 577, 3363, 19601, 114243, …
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A038322 → basis A007318 → θ+1
1 · 19, 16 · 361, 608, 192 · 6859, 17328, 10944, 2048 … Σ 1, 35, 1161, 37179, 1161297, 35606547, …
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A113214 → Veronese 2 (j=1) → x ↦ x+1
1 · 3 · 3, 1 · 5, 5 … Σ 0, 1, 3, 4, 10, 22, 42, 92, …
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A038320 → x ↦ x+1 → D+1
493, 276, 36 · 10115, 8874, 2484, 216 … Σ 805, 21689, 571849, 14831573, …
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A038321 → basis A013609 → x ↦ x/(x+1)
324, 1152, 1024 · 5832, 31104, 55296, 32768 … Σ 2500, 125000, 6250000, 312500000, …
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A137320 → reverse → x ↦ x+1
1 · 2 · 6, 2 · 24, 20, 4 … Σ 1, 2, 8, 48, 384, 3840, 46080, 645120, …
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A322944 → x ↦ x+1
1 · 2, 1 · 9, 8, 1 · 60, 71, 18, 1 … Σ 1, 3, 18, 150, 1560, 19272, 274416, …
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A128100 → basis A013610 → x ↦ x+1
1 · 1 · 6, 3 · 11, 6 … Σ 1, 1, 9, 17, 89, 225, 937, 2737, …
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A132812 → basis A007318 → reverse
1 · 2, 4 · 3, 15, 15 · 4, 36, 84, 56 … Σ 1, 6, 33, 180, 985, 5418, 29953, …
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A145596 → x ↦ x+1 → D+1
1 · 6, 2 · 28, 20, 3 · 120, 136, 44, 4 … Σ 1, 8, 51, 304, 1765, 10128, 57799, …