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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A034870 → basis A013609 → D
8, 8 · 64, 192, 192, 64 … Σ 0, 16, 512, 12288, 262144, 5242880, …
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A257617 → θ+1
1 · 2, 4 · 4, 72, 12 · 8, 776, 1164, 32 … Σ 1, 6, 88, 1980, 59400, 2217600, …
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basis-map coefficients of A297191, shifted → basis A013610 → θ+1
1 · 1, 12, 27 · 1, 24, 162, 432, 405 … Σ 1, 40, 1024, 22528, 458752, 8912896, …
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A199881 → D+1
2, 1 · 1, 3, 1 · 1, 5, 5, 1 · 0, 4, 11, 7, 1 … Σ 3, 5, 12, 23, 45, 84, 155, 281, …
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A013609 → Veronese 2 (j=1) → D+1
2 · 4 · 14, 8 · 40, 32 … Σ 0, 2, 4, 22, 72, 266, 908, 3102, …
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A128099 → θ+1
1 · 1 · 1, 4 · 1, 8 … Σ 1, 1, 5, 9, 25, 53, 125, 273, …
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A209599 → D+1
1 · 2 · 4, 1 · 8, 3 … Σ 1, 2, 5, 11, 25, 55, 121, 263, …
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A013610 → (θ+1)²
1 · 1, 12 · 1, 24, 81 · 1, 36, 243, 432 … Σ 1, 13, 106, 712, 4288, 24064, 128512, …
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A008288 → basis A013611 → x ↦ x/(x+1)
1 · 2, 6 · 5, 30, 41 · 12, 108, 308, 276 … Σ 1, 8, 76, 704, 6544, 60800, 564928, …
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A269944 → basis A013609 → (θ+1)²
1 · 1, 8 · 2, 24, 36 · 10, 136, 288, 128 … Σ 1, 9, 62, 562, 7736, 159560, 4714688, …
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A038320 → basis A013611 → reverse
576, 816, 289 · 13824, 29376, 20808, 4913 … Σ 1681, 68921, 2825761, 115856201, …
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A026729 → Veronese 2 (j=1) → x ↦ x+1
1 · 1 · 1, 1 · 3, 3 … Σ 0, 1, 1, 2, 6, 10, 22, 48, …
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A257612 → (θ+1)²
1 · 2, 8 · 4, 96, 36 · 8, 736, 1656, 128 … Σ 1, 10, 136, 2528, 57856, 1566720, …
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A038330 → (θ+1)²
144, 384, 144 · 1728, 6912, 5184, 1024 … Σ 672, 14848, 311296, 6291456, 123731968, …
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A114687 → basis A013611 → D+1
1 · 11, 8 · 67, 184, 64 · 381, 1872, 2304, 512 … Σ 1, 19, 315, 5069, 81381, 1313271, …
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basis-map coefficients of A030528, shifted → (θ+1)² → x ↦ x+1
1 · -4, -4 · 9, 18, 9 · -16, -48, -48, -16 … Σ 1, -8, 36, -128, 400, -1152, 3136, …
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A048993 → reverse
1 · 1 · 1, 1 · 1, 3, 1 … Σ 1, 1, 2, 5, 15, 52, 203, 877, …
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A038331 → x ↦ x+1 → D+1
459, 220, 25 · 9248, 6885, 1650, 125 … Σ 704, 17908, 447216, 11010032, …
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A142963 → θ+1
1 · 1, 4 · 1, 20, 12 · 1, 60, 216, 32 … Σ 1, 5, 33, 309, 3633, 50661, 816993, …
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basis-map coefficients of A026729 → D → basis A007318
1 · 2, 2 · 3, 6, 3 · 4, 12, 12, 4 … Σ 0, 1, 4, 12, 32, 80, 192, 448, …
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A125662 → basis A013611 → D+1
1 · 8, 4 · 47, 64, 16 · 240, 568, 384, 64 … Σ 1, 12, 127, 1256, 11905, 109604, …
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A035607 → basis A013609 → D+1
1 · 1 · 7, 4 · 23, 32, 8 … Σ 1, 1, 11, 63, 361, 1985, 10659, 56191, …
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A347823 → reverse → x ↦ x+1
1 · 5, 2 · 16, 14, 3 · 44, 60, 27, 4 … Σ 1, 7, 33, 135, 513, 1863, 6561, 22599, …
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reversed rows → x ↦ x+1
1 · 3, 2 · 4, 3 · 7, 8, 2 … Σ 1, 5, 7, 17, 31, 65, 127, 257, …
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A033282 → basis A013610 → θ+1
1 · 3, 12 · 11, 90, 135 · 45, 558, 1701, 1512 … Σ 1, 15, 236, 3816, 62819, 1047309, …
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A003506 → Veronese 2 → D+1
1 · 2 · 6, 3 · 16, 12 … Σ 0, 1, 2, 9, 28, 80, 216, 560, …
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A074246: Triangle of coefficients, read by rows, where the n-th row forms the polynomial P(n,x) = {Sum_{k=1..n} 1/(k+x)}*{Product_{k=1..n} (k+x)}.
1 · 3, 2 · 11, 12, 3 · 50, 70, 30, 4 … Σ 1, 5, 26, 154, 1044, 8028, 69264, …
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A152830 → x ↦ x+1
1 · 1 · 1 · 1 … Σ 1, 1, 1, 1, 3, 3, 3, 3, …
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A038323 → reverse
1 · 9, 11 · 81, 198, 121 · 729, 2673, 3267, 1331 … Σ 1, 20, 400, 8000, 160000, 3200000, …
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A136572 → basis A013611 → D+1
1 · 5, 4 · 18, 80, 32 · 78, 648, 1440, 384 … Σ 1, 9, 130, 2550, 63000, 1875000, …
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A107131 → θ+1
1 · 0, 2 · 0, 2, 3 · 0, 0, 9, 4 … Σ 1, 2, 5, 13, 35, 96, 267, 750, …
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A099040 → basis A013609 → x ↦ x/(x+1)
1 · 2, 6 · 6, 32, 42 · 16, 128, 336, 288 … Σ 1, 8, 80, 768, 7424, 71680, 692224, …
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A133087 → D+1
1 · 3, 1 · 3, 4, 1 · 7, 13, 7, 1 … Σ 1, 4, 8, 28, 48, 160, 256, 832, …
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A038277 → x ↦ x/(x+1) → D+1
1 · 25, 18 · 301, 900, 324 · 2989, 16254, 24300, 5832 … Σ 1, 43, 1525, 49375, 1515625, 44921875, …
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A013609: Triangle of coefficients in expansion of (1+2*x)^n.
1 · 1, 2 · 1, 4, 4 · 1, 6, 12, 8 … Σ 1, 3, 9, 27, 81, 243, 729, 2187, …
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A152842: Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A…
1 · 1, 3 · 1, 4, 3 · 1, 7, 15, 9 … Σ 1, 4, 8, 32, 64, 256, 512, 2048, …
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A137320 → Veronese 2 (j=1) → (θ+1)²
2 · 2 · 4, 32 · 12, 192 … Σ 0, 2, 2, 36, 204, 1456, 11760, 105856, …
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A023531 → Veronese 2 → θ+1
1 · 0, 2 · 0, 0, 3 · 0, 0, 0, 4 … Σ 1, 0, 2, 0, 3, 0, 4, 0, …
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A391533 → Veronese 2 (j=1) → x ↦ x+1
4 · 9, 1 · 18, 6 · 36, 21, 1 … Σ 4, 10, 24, 58, 140, 338, 816, 1970, …
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A013611 → basis A007318 → (θ+1)²
1 · 5, 16 · 25, 160, 144 · 125, 1200, 2160, 1024 … Σ 1, 21, 329, 4509, 57105, 685989, …
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A272866 → basis A013609 → (θ+1)²
1 · 5, 40, 36 · 25, 400, 1260, 1280, 400 … Σ 1, 81, 3365, 110941, 3234921, 87321929, …
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A038328 → reverse
4, 48, 144 · 8, 144, 864, 1728 … Σ 196, 2744, 38416, 537824, 7529536, …
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A027467 → x ↦ x+1 → reverse
1 · 1, 16 · 1, 32, 256 · 1, 48, 768, 4096 … Σ 1, 17, 289, 4913, 83521, 1419857, …
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A103633 → Veronese 2 → x ↦ x+1
1 · 1, 1 · 1, 1 · 2, 3, 1 … Σ 1, 0, 2, 2, 6, 8, 20, 28, …
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basis-map coefficients of A021012 → Veronese 2 → θ+1
1 · 0, 4 · 0, 0, 72 · 0, 0, 0, 2880 … Σ 1, 0, 4, 0, 72, 0, 2880, 0, …
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A028338 → basis A013611 → D
4 · 24, 32 · 176, 384, 192 · 1600, 4480, 3840, 1024 … Σ 0, 4, 56, 752, 10944, 176256, 3142656, …
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A178820 → Veronese 2 → θ+1
1 · 4 · 10, 20 · 20, 120 … Σ 1, 4, 30, 140, 560, 2016, 6720, 21120, …
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A026374 → D+1 → reverse
1 · 1, 2 · 1, 5, 4 · 1, 7, 12, 5 … Σ 1, 3, 10, 25, 75, 175, 500, 1125, …
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A144441: Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
1 · 1, 1 · 1, 14, 1 · 1, 83, 83, 1 … Σ 1, 2, 16, 168, 2416, 44160, 981184, …
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A086645 → basis A013610 → D+1
1 · 5, 3 · 32, 42, 9 · 176, 468, 243, 27 … Σ 1, 8, 83, 914, 9845, 103340, 1062887, …