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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A257609 → D+1
1 · 4, 2 · 20, 24, 4 · 96, 264, 112, 8 … Σ 1, 6, 48, 480, 5760, 80640, 1290240, …
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A026729 → basis A013609 → D+1
1 · 3, 2 · 8, 14, 4 · 17, 54, 44, 8 … Σ 1, 5, 26, 123, 561, 2484, 10773, 45981, …
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A130534 → Veronese 2 → (θ+1)²
1 · 1 · 2, 4 · 6, 24 … Σ 1, 1, 6, 30, 173, 1155, 8807, 75656, …
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A104029 → D
1 · 5, 2 · 13, 18, 3 · 26, 70, 42, 4 … Σ 0, 1, 7, 34, 142, 547, 2005, 7108, …
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A109692 → basis A013610 → x ↦ x/(x+1)
1 · 2, 5 · 8, 46, 65 · 48, 444, 1356, 1365 … Σ 1, 7, 119, 3213, 118881, 5587407, …
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A152198 → basis A013611 → D
4 · 4 · 16, 32 · 16, 32 … Σ 0, 0, 4, 4, 48, 48, 432, 432, …
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A159834: Coefficient array of Hermite_H(n, (x-1)/sqrt(2))/(sqrt(2))^n.
A066325 → x ↦ x−1 (shift -1)1 · -1, 1 · 0, -2, 1 · 2, 0, -3, 1 … Σ 1, 0, -1, 0, 3, 0, -15, 0, …
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A038336 → θ+1
144, 480, 300 · 1728, 8640, 10800, 4000 … Σ 924, 25168, 660176, 16866432, …
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A063967 → basis A007318 → D+1
1 · 3, 1 · 11, 7, 1 · 36, 36, 11, 1 … Σ 1, 4, 19, 84, 366, 1566, 6615, 27648, …
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A038221 → basis A013610 → D+1
1 · 15, 9 · 144, 270, 81 · 1188, 3888, 3645, 729 … Σ 1, 24, 495, 9450, 172125, 3037500, …
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A272866 → basis A007318 → (θ+1)²
1 · 5, 20, 9 · 25, 200, 315, 160, 25 … Σ 1, 34, 725, 12914, 208241, 3149146, …
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A103327 → basis A013611 → D
4 · 48, 32 · 320, 768, 192 · 1792, 7680, 7680, 1024 … Σ 0, 4, 80, 1280, 18176, 240640, 3047424, …
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A207607 → basis A013611 → θ+1
1 · 3, 16 · 8, 72, 96 · 21, 264, 720, 512 … Σ 1, 19, 176, 1517, 12527, 100456, …
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A144698 → (θ+1)²
1 · 1, 16 · 1, 52, 144 · 1, 128, 1017, 1024 … Σ 1, 17, 197, 2170, 24600, 294420, …
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A253284 → (θ+1)²
1 · 2, 8 · 6, 72, 108 · 24, 576, 2160, 1920 … Σ 1, 10, 186, 4680, 146520, 5466960, …
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A060187 → D
1 · 6, 2 · 23, 46, 3 · 76, 460, 228, 4 … Σ 0, 1, 8, 72, 768, 9600, 138240, …
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A008306: Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).
1 · 2 · 6, 3 · 24, 20 … Σ 1, 2, 9, 44, 265, 1854, 14833, 133496, …
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A249252 → θ+1
1 · 1, 8 · 1, 16, 24 · 1, 24, 96, 64 … Σ 1, 9, 41, 185, 793, 3305, 13481, 54105, …
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A257618 → (θ+1)²
1 · 2, 8 · 4, 160, 36 · 8, 1888, 4248, 128 … Σ 1, 10, 200, 6272, 251904, 12295680, …
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A081579 → (θ+1)²
1 · 1, 4 · 1, 24, 9 · 1, 44, 99, 16 … Σ 1, 5, 34, 160, 760, 3296, 13920, 56704, …
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A114656 → basis A007318 → D
1 · 8, 2 · 51, 30, 3 · 304, 292, 72, 4 … Σ 0, 1, 10, 84, 672, 5280, 41184, 320320, …
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A034870 → D
2, 2 · 4, 12, 12, 4 · 6, 30, 60, 60, 30, 6 … Σ 0, 4, 32, 192, 1024, 5120, 24576, …
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A139548 → x ↦ x+1 → reverse
1 · 2, 8, 8 · 4, 32, 96, 128, 64 … Σ 1, 18, 324, 5832, 104976, 1889568, …
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A108838 → x ↦ x+1 → D+1
2 · 7, 2 · 26, 16, 2 · 98, 98, 27, 2 … Σ 2, 9, 44, 225, 1182, 6321, 34232, …
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A104684 → (θ+1)²
1 · 2, 4 · 6, 24, 9 · 20, 120, 108, 16 … Σ 1, 6, 39, 264, 1785, 11958, 79303, …
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A285061 → x ↦ x/(x+1)
1 · 1, 5 · 1, 26, 41 · 1, 127, 491, 429 … Σ 1, 6, 68, 1048, 19472, 420960, …
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A013620 → x ↦ x/(x+1)
1 · 2, 5 · 4, 20, 25 · 8, 60, 150, 125 … Σ 1, 7, 49, 343, 2401, 16807, 117649, …
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A038321 → basis A013610 → D
756, 882 · 20412, 47628, 27783 … Σ 1638, 95823, 4982796, 242911305, …
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A197208 → θ+1
30, 240, 90 · 60, 1200, 1800, 240 … Σ 360, 3300, 28980, 248136, 2090592, …
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A225467 → D
4 · 40, 32 · 316, 672, 192 · 2320, 9920, 6912, 1024 … Σ 0, 4, 72, 1180, 20176, 368884, 7232408, …
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A085881 → basis A013611 → basis A007318
1 · 6, 4 · 108, 144, 48 · 3240, 6480, 4320, 960 … Σ 1, 10, 300, 15000, 1050000, 94500000, …
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A103327 → basis A013610 → (θ+1)²
1 · 4, 12 · 16, 144, 81 · 64, 960, 1944, 432 … Σ 1, 16, 241, 3400, 44377, 545992, …
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A178822 → basis A013609 → (θ+1)²
1 · 12, 48 · 84, 672, 756 · 448, 5376, 12096, 7168 … Σ 1, 60, 1512, 25088, 322560, 3483648, …
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A109692 → basis A013610 → (θ+1)²
1 · 2, 12 · 8, 120, 243 · 48, 1200, 5508, 6480 … Σ 1, 14, 371, 13236, 584877, 30684018, …
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A116414 → reverse → x ↦ x/(x+1)
1 · 1, 5 · 1, 10, 22 · 1, 15, 69, 95 … Σ 1, 6, 33, 180, 981, 5346, 29133, …
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A172431 → basis A013610 → D+1
1 · 9, 6 · 38, 84, 27 · 135, 510, 522, 108 … Σ 1, 15, 149, 1275, 10177, 78183, 586349, …
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A142458 → D
1 · 8, 2 · 39, 78, 3 · 166, 1092, 498, 4 … Σ 0, 1, 10, 120, 1760, 30800, 628320, …
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basis-map coefficients of A112227 → θ+1 → x ↦ x+1
3, 6, 3 · 4, 12, 12, 4 · 5, 20, 30, 20, 5 … Σ 12, 32, 80, 192, 448, 1024, 2304, 5120, …
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A038220 → basis A013611 → D
8 · 80, 128 · 600, 1920, 1536 … Σ 0, 8, 208, 4056, 70304, 1142440, …
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A108838 → reverse → D+1
2 · 5, 3 · 10, 16, 4 · 17, 55, 35, 5 … Σ 2, 8, 30, 112, 420, 1584, 6006, 22880, …
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A200536 → basis A007318 → basis A007318
1 · 15, 11, 2 · 225, 330, 181, 44, 4 … Σ 1, 28, 784, 21952, 614656, 17210368, …
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A075500 → x ↦ x+1
41, 17, 1 · 331, 238, 33, 1 · 3176, 3279, 781, 54, 1 … Σ 59, 603, 7291, 101107, 1571299, …
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A075497 → reverse → (θ+1)²
1 · 1, 8 · 1, 24, 36 · 1, 48, 252, 128 … Σ 1, 9, 61, 429, 3301, 27533, 246121, …
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A027466: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).
1 · 7, 1 · 49, 14, 1 · 343, 147, 21, 1 … Σ 1, 8, 64, 512, 4096, 32768, 262144, …
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image under θ+1 → (θ+1)² → D+1
1 · 8, 8 · 0, 54, 27 · 0, 0, 192, 64 … Σ 1, 16, 81, 256, 625, 1296, 2401, 4096, …
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A113953 → (θ+1)²
1 · 0, 4 · 0, 8, 9 · 0, 0, 36, 16 … Σ 1, 4, 17, 52, 157, 428, 1137, 2892, …
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A013612 → basis A013610 → θ+1
36, 360, 675 · 216, 3240, 12150, 13500 … Σ 1071, 29106, 750141, 18670176, …
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A131431 → x ↦ x+1 → (θ+1)²
1 · 4, 16 · 7, 56, 63 · 10, 120, 270, 160 … Σ 1, 20, 126, 560, 2080, 6912, 21280, …
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A059473 → D
2 · 10, 8 · 32, 64, 24 · 88, 296, 264, 64 … Σ 0, 2, 18, 120, 712, 3960, 21144, …
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A334218 → D → x ↦ x+1
2 · 9, 6 · 48, 56, 12 · 300, 500, 225, 20 … Σ 0, 0, 2, 15, 116, 1045, 10854, 127883, …