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_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_of_power_multiple, shift 1 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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A063007 → x ↦ x+1 → (θ+1)²
1 · 3, 8 · 13, 72, 54 · 63, 528, 810, 320 … Σ 1, 11, 139, 1721, 20761, 244955, …
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A228955 → (θ+1)²
1 · 1, 4 · 2, 24 · 6, 144, 54 … Σ 1, 5, 26, 204, 2064, 25440, 368640, …
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A038333 → (θ+1)²
144, 672, 441 · 1728, 12096, 15876, 5488 … Σ 1257, 35188, 918745, 22881624, …
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A082985 → x ↦ x+1 → (θ+1)²
1 · 4, 12 · 11, 60, 45 · 29, 224, 315, 112 … Σ 1, 16, 116, 680, 3652, 18712, 92964, …
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A086645 → Veronese 2 → D+1
1 · 1 · 2, 1 · 16, 15 … Σ 1, 1, 3, 31, 144, 556, 2480, 11376, …
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A280580 → (θ+1)²
1 · 1, 4 · 2, 24, 9 · 5, 120, 135, 16 … Σ 1, 5, 35, 276, 2307, 19893, 174685, …
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A391045 → x ↦ x+1
1 · 2, 1 · 6, 6, 1 · 21, 32, 13, 1 … Σ 1, 3, 13, 67, 377, 2243, 13877, 88371, …
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chebyshev_t → D
1 · 0, 4 · -3, 0, 12 · 0, -16, 0, 32 … Σ 0, 1, 4, 9, 16, 25, 36, 49, …
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A232535 → D+1
1 · 3, 2 · 9, 14, 3 · 19, 68, 37, 4 … Σ 1, 5, 26, 128, 608, 2816, 12800, 57344, …
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A249139 → basis A013609 → θ+1
1 · 4, 4 · 7, 8 · 19, 36, 12 … Σ 1, 8, 15, 67, 164, 581, 1593, 5100, …
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A054142 → Veronese 2 (j=1) → D
1 · 10 · 35, 2 · 84, 42 … Σ 0, 0, 0, 1, 10, 37, 126, 420, …
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A206306 → D
1 · 3, 2 · 7, 12, 3 · 15, 46, 27, 4 … Σ 0, 1, 5, 22, 92, 372, 1468, 5688, …
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basis-map coefficients of A371080, shifted → basis A013609 → reverse
1 · 2, 1 · 4, 10, 4 · 8, 48, 78, 28 … Σ 1, 3, 18, 162, 1944, 29160, 524880, …
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A257614 → (θ+1)²
1 · 2, 8 · 4, 112, 36 · 8, 976, 2196, 128 … Σ 1, 10, 152, 3308, 90000, 2926224, …
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A099093 → θ+1 → D
6 · 6, 54 · 0, 108, 324 · 0, 54, 972, 1620 … Σ 0, 6, 60, 432, 2646, 14742, 77112, …
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narayana → basis A007318
1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … Σ 1, 3, 11, 45, 197, 903, 4279, 20793, …
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A038335 → x ↦ x+1 → D+1
1 · 30, 9 · 819, 540, 81 · 21168, 22113, 7290, 729 … Σ 1, 39, 1440, 51300, 1782000, 60750000, …
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A143491 → basis A007318 → reverse
1, 7, 12 · 1, 12, 47, 60 · 1, 18, 119, 342, 360 … Σ 20, 120, 840, 6720, 60480, 604800, …
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A078812 → basis A013609 → reverse
1 · 2, 3 · 4, 12, 8 · 8, 36, 50, 21 … Σ 1, 5, 24, 115, 551, 2640, 12649, 60605, …
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basis-map coefficients of A112227 → basis A013611 → reverse
16, 8, 1 · 64, 48, 12, 1 · 256, 256, 96, 16, 1 … Σ 25, 125, 625, 3125, 15625, 78125, …
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A013618 → reverse
121, 22, 1 · 1331, 363, 33, 1 … Σ 144, 1728, 20736, 248832, 2985984, …
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A062264 → x ↦ x+1 → θ+1
1 · 6, 10 · 28, 84, 45 · 120, 504, 504, 140 … Σ 1, 16, 157, 1268, 9281, 64168, 427885, …
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A038330 → basis A013609 → x ↦ x+1
576, 384, 64 · 13824, 13824, 4608, 512 … Σ 1024, 32768, 1048576, 33554432, …
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A290315 → (θ+1)²
1 · 1, 8 · 1, 64, 108 · 1, 264, 2556, 1920 … Σ 1, 9, 173, 4741, 163977, 6794161, …
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A038322 → D
8 · 176, 128 · 2904, 4224, 1536 … Σ 0, 8, 304, 8664, 219488, 5212840, …
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A038337 → basis A013609 → reverse
484, 1012, 529 · 10648, 33396, 34914, 12167 … Σ 2025, 91125, 4100625, 184528125, …
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A099040 → θ+1 → D
4 · 4, 24 · 0, 48, 96 · 0, 24, 288, 320 … Σ 0, 4, 28, 144, 632, 2528, 9504, 34176, …
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A038320 → (θ+1)²
121, 528, 324 · 1331, 8712, 10692, 3456 … Σ 973, 24191, 562105, 12474107, …
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A056241 → (θ+1)²
1 · 1, 4 · 1, 12, 9 · 1, 24, 54, 16 … Σ 1, 5, 22, 95, 397, 1597, 6206, 23431, …
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A391403 → x ↦ x+1 → θ+1
1 · 1 · 3, 4 · 7, 12 … Σ 1, 1, 7, 19, 85, 301, 1211, 4607, …
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A131431 → (θ+1)² → D
16 · 0, 126 · 0, 0, 480 · 0, 0, 0, 1300 … Σ 0, 16, 126, 480, 1300, 2880, 5586, …
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A038335 → (θ+1)²
1 · 12, 36 · 144, 864, 729 … Σ 1, 48, 1737, 55188, 1623321, 45341856, …
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A159854 → Veronese 2 → θ+1
0, 2 · 0, 2 · 0, 2, 3 · 0, 2, 9 … Σ 2, 2, 5, 11, 24, 52, 112, 240, …
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A391045 → θ+1 → D+1
1 · 3, 2 · 9, 14, 3 · 19, 78, 42, 4 … Σ 1, 5, 26, 143, 788, 4332, 23735, …
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A330798 → x ↦ x+1
1 · 4, 2 · 30, 33, 9 · 264, 452, 256, 48 … Σ 1, 6, 72, 1020, 15510, 245124, 3968538, …
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A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
1 · 0, 1 · 0, 0, 1 · 0, 0, 0, 1 … Σ 1, 1, 1, 1, 1, 1, 1, 1, …
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A130534 → Veronese 2 (j=1) → θ+1
1 · 3 · 11, 2 · 50, 20 … Σ 0, 1, 3, 13, 70, 447, 3297, 27576, …
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A038331 → x ↦ x+1 → (θ+1)²
289, 680, 225 · 4913, 17340, 11475, 2000 … Σ 1194, 35728, 1018336, 28046832, …
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A391533 → x ↦ x+1 → D
6, 2 · 21, 16, 3 · 60, 74, 30, 4 … Σ 8, 40, 168, 648, 2376, 8424, 29160, …
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A038320 → x ↦ x+1
289, 204, 36 · 4913, 5202, 1836, 216 … Σ 529, 12167, 279841, 6436343, 148035889, …
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A055151 → basis A007318 → (θ+1)²
1 · 1 · 2, 4 · 4, 12 … Σ 1, 1, 6, 16, 67, 231, 896, 3326, …
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A094816 → basis A013611 → D
4 · 20, 32 · 92, 288, 192 · 464, 2080, 2688, 1024 … Σ 0, 4, 52, 572, 6256, 71156, 855220, …
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A008292 → x ↦ x+1 → (θ+1)²
1 · 2, 4 · 6, 24, 9 · 24, 144, 126, 16 … Σ 1, 6, 39, 310, 2935, 32186, 400855, …
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A269944 → basis A013609 → reverse
1 · 2, 1 · 4, 6, 2 · 8, 32, 34, 10 … Σ 1, 3, 12, 84, 1008, 19152, 536256, …
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A034870 → Veronese 2 (j=1) → D+1
2 · 8, 4 · 26, 32, 6 · 64, 168, 80, 8 … Σ 0, 2, 12, 64, 320, 1536, 7168, 32768, …
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A051523: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 9.
1 · -10, 1 · 110, -21, 1 · -1320, 362, -33, 1 … Σ 1, -9, 90, -990, 11880, -154440, …
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A008278 → x ↦ x+1 (shift +1) → x ↦ x+1 → D
1 · 7, 2 · 46, 26, 3 · 322, 278, 69, 4 … Σ 0, 1, 9, 75, 673, 6588, 69792, 795181, …
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A116412 → basis A013609 → reverse
1 · 2, 4 · 4, 16, 13 · 8, 48, 84, 43 … Σ 1, 6, 33, 183, 1014, 5619, 31137, …
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A038338 → x ↦ x+1 → θ+1
576, 1152, 432 · 13824, 41472, 31104, 6912 … Σ 2160, 93312, 3919104, 161243136, …
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A008278 → x ↦ x+1 (shift +1) → reverse → x ↦ x/(x+1)
1 · 1, 3 · 1, 7, 11 · 1, 13, 46, 49 … Σ 1, 4, 19, 109, 742, 5815, 51193, …