preserver_image / euler_polynomial, shift 0
Image under a real-rootedness preserver. All operations
Parameters (20)
- all
- derivative_polynomial, shift 0 422 proof steps, 410 sequences
- euler_polynomial, shift 0 367 proof steps, 352 sequences
- affine, shift 0 333 proof steps, 318 sequences
- reversal, shift 0 169 proof steps, 164 sequences
- mobius, shift 0 160 proof steps, 160 sequences
- hadamard_multiplier, shift 0 5 proof steps, 5 sequences
- affine, shift -1 5 proof steps, 5 sequences
- affine, shift 1 4 proof steps, 4 sequences
- reversal, shift 1 4 proof steps, 4 sequences
- hadamard_multiplier, shift 2 3 proof steps, 3 sequences
- euler_polynomial, shift 1 2 proof steps, 2 sequences
- hadamard_multiplier, shift -1 2 proof steps, 2 sequences
- affine, shift 2 2 proof steps, 2 sequences
- derivative_of_power_multiple, shift 0 1 proof steps, 1 sequences
- derivative_of_power_multiple, shift 1 1 proof steps, 1 sequences
- derivative_polynomial, shift -1 1 proof steps, 1 sequences
- derivative_polynomial, shift 2 1 proof steps, 1 sequences
- euler_polynomial, shift 2 1 proof steps, 1 sequences
- hadamard_multiplier, shift 1 1 proof steps, 1 sequences
- reversal, shift -1 1 proof steps, 1 sequences
352 sequences
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A132393 → θ+1
1 · 0, 2 · 0, 2, 3 · 0, 4, 9, 4 … row sums 1, 2, 5, 17, 74, 394, 2484, 18108, …
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A122542 → θ+1
1 · 0, 2 · 0, 4, 3 · 0, 4, 12, 4 … row sums 1, 2, 7, 20, 57, 158, 431, 1160, …
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A131689 → (θ+1)²
1 · 0, 4 · 0, 4, 18 · 0, 4, 54, 96 … row sums 1, 4, 22, 154, 1306, 12994, 148282, …
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A131689 → θ+1
1 · 0, 2 · 0, 2, 6 · 0, 2, 18, 24 … row sums 1, 2, 8, 44, 308, 2612, 25988, 296564, …
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A008459 → θ+1
1 · 1, 2 · 1, 8, 3 · 1, 18, 27, 4 … row sums 1, 3, 12, 50, 210, 882, 3696, 15444, …
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A028326 → (θ+1)²
2 · 2, 8 · 2, 16, 18 · 2, 24, 54, 32 … row sums 2, 10, 36, 112, 320, 864, 2240, 5632, …
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basis-map coefficients of A030528, shifted → θ+1
1 · 0, -2 · 0, 0, 3 · 0, 0, 0, -4 … row sums 1, -2, 3, -4, 5, -6, 7, -8, …
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A008278 → (θ+1)²
1 · 1, 4 · 1, 12, 9 · 1, 24, 63, 16 … row sums 1, 5, 22, 104, 531, 2897, 16787, …
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A137320 → (θ+1)²
1 · 0, 8 · 0, 8, 36 · 0, 16, 108, 128 … row sums 1, 8, 44, 252, 1612, 11624, 94040, …
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A111125 → (θ+1)²
1 · 3, 4 · 5, 20, 9 · 7, 56, 63, 16 … row sums 1, 7, 34, 142, 541, 1939, 6653, 22091, …
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A023531 → 2D+1+2n → (θ+1)²
1 · 2, 12 · 0, 16, 45 · 0, 0, 54, 112 … row sums 1, 14, 61, 166, 353, 646, 1069, 1646, …
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A100861 → reverse → (θ+1)²
1 · 0, 4 · 0, 4, 9 · 0, 0, 27, 16 … row sums 1, 4, 13, 43, 148, 526, 1954, 7498, …
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A026729 → (θ+1)²
1 · 0, 4 · 0, 4, 9 · 0, 0, 18, 16 … row sums 1, 4, 13, 34, 82, 184, 395, 818, …
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A391533 → θ+1
4, 8, 3 · 4, 16, 15, 4 · 4, 24, 39, 24, 5 … row sums 15, 39, 96, 228, 528, 1200, 2688, 5952, …
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A111125 → θ+1
1 · 3, 2 · 5, 10, 3 · 7, 28, 21, 4 … row sums 1, 5, 18, 60, 191, 589, 1775, 5257, …
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A075497 → (θ+1)²
1 · 2, 4 · 4, 24, 9 · 8, 112, 108, 16 … row sums 1, 6, 37, 244, 1741, 13442, 111833, …
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A391533 → (θ+1)²
4, 16, 9 · 4, 32, 45, 16 · 4, 48, 117, 96, 25 … row sums 29, 97, 290, 808, 2144, 5488, 13664, …
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A102365 → θ+1
1 · 1 · 1, 2 · 1, 10, 3 … row sums 1, 1, 3, 14, 86, 657, 5995, 63541, …
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A144438 → θ+1
1 · 1, 2 · 1, 10, 3 · 1, 28, 42, 4 … row sums 1, 3, 14, 75, 471, 3402, 27844, 254970, …
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A125662 → θ+1
1 · 3, 2 · 8, 12, 3 · 21, 50, 27, 4 … row sums 1, 5, 23, 102, 441, 1871, 7823, 32332, …
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A144438 → (θ+1)²
1 · 1, 4 · 1, 20, 9 · 1, 56, 126, 16 … row sums 1, 5, 30, 199, 1487, 12450, 115866, …
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A008292 → γ (scale 1/4) → θ+1
1 · 2 · 4, 4 · 8, 32 … row sums 1, 2, 8, 40, 240, 1680, 13440, 120960, …
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A011973 → ⊙ C(n,k) → θ+1
1 · 1 · 1, 4 · 1, 12 … row sums 1, 1, 5, 13, 43, 131, 411, 1275, …
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A039756 → θ+1
1 · 1, 2 · 1, 8, 3 · 1, 18, 39, 4 … row sums 1, 3, 12, 62, 372, 2492, 18384, 147800, …
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A126216 → (θ+1)²
1 · 2, 4 · 5, 20, 9 · 14, 84, 81, 16 … row sums 1, 6, 34, 195, 1131, 6608, 38788, …
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A136572 → (θ+1)²
1 · 0, 4 · 0, 0, 18 · 0, 0, 0, 96 … row sums 1, 4, 18, 96, 600, 4320, 35280, 322560, …
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A110441 → θ+1
1 · 3, 2 · 7, 12, 3 · 15, 46, 27, 4 … row sums 1, 5, 22, 92, 372, 1468, 5688, 21728, …
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A127733: Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
A023531 → (θ+1)²1 · 0, 4 · 0, 0, 9 · 0, 0, 0, 16 … row sums 1, 4, 9, 16, 25, 36, 49, 64, …
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A367270 → θ+1
1 · 1 · 1, 4 · 1, 12, 9 … row sums 1, 1, 5, 22, 95, 406, 1722, 7260, …
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A085880 → (θ+1)²
1 · 1, 4 · 2, 16, 18 · 5, 60, 135, 80 … row sums 1, 5, 36, 280, 2240, 18144, 147840, …
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A023531 → (1/2)D²+2D+1 → (θ+1)²
1 · 2, 4 · 1, 16, 9 · 0, 12, 54, 16 … row sums 1, 6, 26, 82, 207, 446, 856, 1506, …
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A207606 → (θ+1)²
4, 28, 18 · 5, 64, 99, 32 · 6, 120, 324, 240, 50 … row sums 50, 200, 740, 2597, 8770, 28759, 92146, …
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A127648: Triangle read by rows: row n consists of n zeros followed by n+1.
A010054 → reverse1 · 0, 2 · 0, 0, 3 · 0, 0, 0, 4 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
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A054142 → (θ+1)²
1 · 1, 4 · 1, 12, 9 · 1, 20, 54, 16 … row sums 1, 5, 22, 91, 349, 1260, 4349, 14509, …
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A038231 → (θ+1)²
1 · 4, 4 · 16, 32, 9 · 64, 192, 108, 16 … row sums 1, 8, 57, 380, 2425, 15000, 90625, …
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basis-map coefficients of A113214, shifted → (θ+1)²
1 · -1, -4 · 1, 8, 9 · -1, -12, -27, -16 … row sums 1, -5, 18, -56, 160, -432, 1120, -2816, …
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A007318 → D → θ+1
1 · 2, 4 · 3, 12, 9 · 4, 24, 36, 16 … row sums 0, 1, 6, 24, 80, 240, 672, 1792, …
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A178821 → (θ+1)²
1 · 5, 20 · 15, 120, 135 · 35, 420, 945, 560 … row sums 1, 25, 270, 1960, 11200, 54432, 235200, …
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A108838 → (θ+1)²
2 · 3, 8 · 4, 32, 18 · 5, 80, 135, 32 … row sums 2, 11, 54, 252, 1140, 5049, 22022, …
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A178821 → θ+1
1 · 5, 10 · 15, 60, 45 · 35, 210, 315, 140 … row sums 1, 15, 120, 700, 3360, 14112, 53760, …
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A103327 → (θ+1)²
1 · 3, 4 · 5, 40, 9 · 7, 140, 189, 16 … row sums 1, 7, 54, 352, 2080, 11520, 60928, …
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A165253 → θ+1
1 · 1 · 1, 2 · 1, 6, 3 … row sums 1, 1, 3, 10, 32, 99, 299, 887, …
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A114656 → (θ+1)²
1 · 2, 4 · 4, 24, 9 · 8, 96, 108, 16 … row sums 1, 6, 37, 228, 1401, 8578, 52333, …
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A145596 → (θ+1)²
1 · 2, 8 · 3, 32, 27 · 4, 80, 180, 64 … row sums 1, 10, 62, 328, 1605, 7502, 34034, …
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A062145 → (θ+1)²
1 · 1, 16 · 1, 40, 90 · 1, 72, 405, 320 … row sums 1, 17, 131, 798, 4362, 22407, 110539, …
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A008459 → (θ+1)²
1 · 1, 4 · 1, 16, 9 · 1, 36, 81, 16 … row sums 1, 5, 26, 134, 670, 3262, 15540, 72732, …
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A081578 → (θ+1)²
1 · 1, 4 · 1, 20, 9 · 1, 36, 81, 16 … row sums 1, 5, 30, 134, 583, 2355, 9196, 34748, …
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A008277 → θ+1
1 · 1, 2 · 1, 6, 3 · 1, 14, 18, 4 … row sums 0, 1, 3, 10, 37, 151, 674, 3263, …
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A010054 → reverse → (θ+1)²
1 · 0, 12 · 0, 0, 54 · 0, 0, 0, 160 … row sums 1, 12, 54, 160, 375, 756, 1372, 2304, …
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A063007 → (θ+1)²
1 · 1, 8 · 1, 24, 54 · 1, 48, 270, 320 … row sums 1, 9, 79, 639, 4881, 35793, 254647, …