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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A144442 → θ+1
1 · 1, 2 · 1, 34, 3 · 1, 236, 354, 4 … Σ 1, 3, 38, 595, 12423, 323022, 10050356, …
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narayana → D → (θ+1)²
1 · 3, 8 · 6, 48, 27 · 10, 160, 270, 64 … Σ 0, 1, 11, 81, 504, 2850, 15147, 77077, …
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A093375 → D+1
1 · 3, 1 · 7, 6, 1 · 13, 21, 9, 1 … Σ 1, 4, 14, 44, 128, 352, 928, 2368, …
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A099093 → D
3 · 3, 18 · 0, 36, 81 · 0, 18, 243, 324 … Σ 0, 3, 21, 117, 585, 2754, 12474, 54999, …
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A086645 → D → D+1
1 · 8, 2 · 45, 36, 3 · 168, 308, 96, 4 … Σ 0, 1, 10, 84, 576, 3520, 19968, 107520, …
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A172431 → basis A013610 → D
6 · 30, 54 · 114, 396, 324 · 390, 1998, 3240, 1620 … Σ 0, 6, 84, 834, 7248, 58878, 459468, …
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Veronese parent of A098172 → θ+1
1 · 0, 0, 3, 4 · 0, 0, 0, 0, 5, 12, 7 … Σ 1, 7, 24, 68, 176, 432, 1024, 2368, …
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A156992 → basis A013611 → D
8 · 96, 192 · 1152, 4608, 4608 … Σ 0, 8, 288, 10368, 414720, 18662400, …
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A094262 → D+1
1 · 3, 4, 1 · 7, 30, 42, 22, 3 … Σ 1, 8, 104, 1888, 44032, 1254656, …
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A113413 → basis A013610 → D+1
1 · 6, 3 · 25, 36, 9 · 86, 231, 162, 27 … Σ 1, 9, 70, 506, 3494, 23394, 153182, …
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A166972 → D
1 · 10, 2 · 83, 82, 3 · 668, 2220, 366, 4 … Σ 0, 1, 12, 168, 3258, 83184, 2666952, …
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A133087 → x ↦ x+1 → x ↦ x/(x+1)
1 · 3, 4 · 4, 12, 9 · 12, 52, 75, 36 … Σ 1, 7, 25, 175, 625, 4375, 15625, …
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A091044 → basis A007318
1 · 4, 2 · 16, 16, 3 · 64, 96, 40, 4 … Σ 1, 6, 35, 204, 1189, 6930, 40391, …
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A207607 → basis A013610 → D
6 · 27, 36 · 99, 270, 162 · 333, 1368, 1701, 648 … Σ 0, 6, 63, 531, 4050, 29172, 202473, …
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A290319 → basis A013609 → x ↦ x/(x+1)
1 · 2, 4 · 12, 40, 32 · 120, 544, 800, 384 … Σ 1, 6, 84, 1848, 55440, 2106720, …
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A196347 → basis A013609 → D+1
1 · 4, 2 · 24, 32, 8 · 192, 432, 288, 48 … Σ 1, 6, 64, 960, 18432, 430080, 11796480, …
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A144438 → x ↦ x+1
1 · 2, 1 · 7, 7, 1 · 30, 45, 17, 1 … Σ 1, 3, 15, 93, 703, 6285, 64983, 763533, …
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A304236 → basis A007318 → θ+1
1 · 1 · 4, 6 · 7, 12 … Σ 1, 1, 10, 19, 100, 253, 1018, 2935, …
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A013622 → basis A013609 → basis A007318
1 · 18, 10 · 324, 360, 100 · 5832, 9720, 5400, 1000 … Σ 1, 28, 784, 21952, 614656, 17210368, …
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basis-map coefficients of A297191, shifted → basis A013610 → D+1
1 · 7, 24, 9 · 13, 120, 378, 432, 81 … Σ 1, 40, 1024, 22528, 458752, 8912896, …
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A120910 → D → (θ+1)²
3 · 12, 24 · 36, 144, 81 · 96, 576, 648, 192 … Σ 0, 3, 36, 261, 1512, 7695, 35964, …
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A060920 → D+1
1 · 3, 1 · 10, 7, 1 · 33, 38, 12, 1 … Σ 1, 4, 18, 84, 390, 1784, 8042, 35804, …
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A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
1 · 4, 1 · 16, 8, 1 · 64, 48, 12, 1 … Σ 1, 5, 25, 125, 625, 3125, 15625, 78125, …
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A038291: Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.
1 · 9, 1 · 81, 18, 1 · 729, 243, 27, 1 … Σ 1, 10, 100, 1000, 10000, 100000, …
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A207607 → reverse → x ↦ x+1
1 · 3, 1 · 8, 7, 1 · 21, 30, 12, 1 … Σ 1, 4, 16, 64, 256, 1024, 4096, 16384, …
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A085881 → (θ+1)²
1 · 1, 4 · 3, 24, 27 · 15, 180, 405, 240 … Σ 1, 5, 54, 840, 16800, 408240, 11642400, …
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A021010: Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).
A144084 → x ↦ −x (shift -1)1 · -1, 1 · 1, -4, 2 · -1, 9, -18, 6 … Σ 1, 0, -1, -4, -15, -56, -185, -204, …
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A125092 → D
4 · 8, 18 · 12, 54, 48 · 16, 108, 192, 100 … Σ 0, 4, 26, 114, 416, 1360, 4128, 11872, …
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A137320 → reverse
1 · 2 · 4, 2 · 8, 12, 4 … Σ 1, 2, 6, 24, 120, 720, 5040, 40320, …
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A204420 → basis A013610 → x ↦ x/(x+1)
1 · 1, 4 · 9, 54, 72 · 225, 1710, 3960, 2880 … Σ 1, 5, 135, 8775, 1044225, 197358525, …
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A100861 → (θ+1)²
1 · 1 · 1, 4 · 1, 12 … Σ 1, 1, 5, 13, 52, 176, 706, 2710, …
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A122542 → reverse → D
2 · 4, 4 · 6, 16, 6 · 8, 36, 36, 8 … Σ 0, 0, 2, 8, 28, 88, 262, 752, …
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A081579 → γ → D+1
1 · 1 · 5, 4 · 9, 8 … Σ 1, 1, 9, 17, 73, 177, 585, 1553, …
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A152198 → basis A013611 → D+1
1 · 1 · 6, 4 · 6, 4 … Σ 1, 1, 10, 10, 84, 84, 648, 648, …
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type_b_eulerian → D
1 · 6, 2 · 23, 46, 3 · 76, 460, 228, 4 … Σ 0, 1, 8, 72, 768, 9600, 138240, …
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A126216 → D
1 · 5, 2 · 21, 18, 3 · 84, 112, 42, 4 … Σ 0, 1, 7, 42, 242, 1375, 7773, 43876, …
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A038207 → Veronese 2 (j=1) → x ↦ x+1
1 · 4 · 13, 1 · 40, 8 … Σ 0, 1, 4, 14, 48, 164, 560, 1912, …
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A103406 → D+1
1 · 3, 2 · 5, 10, 3 · 7, 24, 21, 4 … Σ 1, 5, 18, 56, 160, 432, 1120, 2816, …
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A178821 → basis A013610 → θ+1
1 · 10, 30 · 60, 360, 405 · 280, 2520, 5670, 3780 … Σ 1, 40, 825, 12250, 148750, 1575000, …
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A290319 → basis A013611 → x ↦ x/(x+1)
1 · 2, 6 · 12, 56, 60 · 120, 728, 1384, 840 … Σ 1, 8, 128, 3072, 98304, 3932160, …
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A156365 → D
8, 8 · 22, 88, 24 · 52, 528, 624, 64 … Σ 16, 134, 1268, 13586, 163560, 2191934, …
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A055151 → basis A007318
1 · 1 · 2, 1 · 4, 3 … Σ 1, 1, 3, 7, 21, 61, 191, 603, …
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A130850 → θ+1
1 · 1, 2 · 2, 6, 3 · 6, 24, 21, 4 … Σ 1, 3, 11, 55, 359, 2891, 27635, 305439, …
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mobius_x_over_x_plus_one of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.
1 · 1, 4, 4 · 1, 10, 34, 48, 24 … Σ 1, 9, 117, 2025, 43821, 1137969, …
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basis-map coefficients of A021012 → basis A008277 → x ↦ x+1
1 · -2, -1 · 10, 10, 2 · -90, -132, -54, -6 … Σ 1, -3, 22, -282, 5448, -145800, …
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A367270 → D → D+1
2 · 12, 6 · 48, 60, 12 · 140, 360, 180, 20 … Σ 0, 0, 2, 18, 120, 700, 3780, 19404, …
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A030528 → basis A013609 → D
2 · 8, 8 · 20, 48, 24 · 44, 168, 192, 64 … Σ 0, 2, 16, 92, 468, 2226, 10152, 44982, …
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A013612 → basis A013610 → D
180, 450 · 1620, 8100, 10125 … Σ 630, 19845, 555660, 14586075, …
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basis-map coefficients of A026729 → basis A013610
1 · 1, 3 · 1, 6, 9 · 1, 9, 27, 27 … Σ 1, 4, 16, 64, 256, 1024, 4096, 16384, …
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A038331 → basis A013610 → D+1
799, 960, 225 · 17918, 35955, 21600, 3375 … Σ 1984, 78848, 3014656, 112197632, …