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
- affine, shift 2 2 proof steps, 2 sequences
- euler_polynomial, shift 1 2 proof steps, 2 sequences
- hadamard_multiplier, shift -1 2 proof steps, 2 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_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
352 sequences
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A011973 → ⊙ C(n,k) → (θ+1)²
1 · 1 · 1, 8 · 1, 24 … row sums 1, 1, 9, 25, 103, 351, 1251, 4299, …
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A114687 → θ+1
1 · 1, 4 · 1, 12, 12 · 1, 24, 72, 32 … row sums 1, 5, 25, 129, 681, 3653, 19825, …
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A178822 → θ+1
1 · 6, 12 · 21, 84, 63 · 56, 336, 504, 224 … row sums 1, 18, 168, 1120, 6048, 28224, 118272, …
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A135838 → θ+1
1 · 2, 4 · 2, 8, 6 · 4, 24, 36, 16 … row sums 1, 6, 16, 80, 192, 896, 2048, 9216, …
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A266213 → θ+1
1 · 1 · 1, 4 · 1, 8, 6 … row sums 1, 1, 5, 15, 45, 129, 361, 991, …
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A103633 → θ+1
1 · 0, 2 · 0, 2, 3 · 0, 0, 3, 4 … row sums 1, 2, 5, 7, 16, 20, 44, 52, …
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A033282 → (θ+1)²
1 · 1, 8 · 1, 20, 45 · 1, 36, 189, 224 … row sums 1, 9, 66, 450, 2955, 18963, 119812, …
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A094638 → θ+1
1 · 1, 2 · 1, 6, 6 · 1, 12, 33, 24 … row sums 1, 3, 13, 70, 446, 3276, 27252, 253296, …
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basis-map coefficients of A112227 → θ+1
0, 0, 3 · 0, 0, 0, 4 · 0, 0, 0, 0, 5 … row sums 3, 4, 5, 6, 7, 8, 9, 10, …
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A114656 → θ+1
1 · 2, 2 · 4, 12, 3 · 8, 48, 36, 4 … row sums 1, 4, 19, 96, 501, 2668, 14407, 78592, …
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A137320 → θ+1
1 · 0, 4 · 0, 4, 12 · 0, 8, 36, 32 … row sums 1, 4, 16, 76, 428, 2808, 21096, 178848, …
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A367270 → (θ+1)²
1 · 1 · 1, 8 · 1, 24, 27 … row sums 1, 1, 9, 52, 275, 1386, 6762, 32208, …
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A062196 → θ+1
1 · 1, 6 · 1, 16, 18 · 1, 30, 90, 40 … row sums 1, 7, 35, 161, 714, 3102, 13299, 56485, …
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A249250 → (θ+1)²
1 · 1, 12 · 1, 24, 27 · 1, 36, 135, 48 … row sums 1, 13, 52, 220, 832, 3028, 10600, …
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A133080 → θ+1
1 · 1, 2 · 0, 0, 3 · 0, 0, 3, 4 … row sums 1, 3, 3, 7, 5, 11, 7, 15, …
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A126351 → (θ+1)²
1 · 1, 8 · 1, 20, 36 · 1, 36, 171, 128 … row sums 1, 9, 57, 336, 1992, 12193, 77725, …
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A100862 → (θ+1)²
1 · 1, 4 · 1, 12, 9 · 1, 24, 54, 16 … row sums 1, 5, 22, 95, 415, 1847, 8410, 39209, …
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A237765: Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
A007318 → ×(−(1/2)n+(1/2)n²)1, 2, 1 · 3, 9, 9, 3 · 6, 24, 36, 24, 6 … row sums 0, 0, 4, 24, 96, 320, 960, 2688, …
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A046716 → θ+1
1 · 1, 2 · 1, 6, 3 · 1, 12, 24, 4 … row sums 1, 3, 10, 41, 209, 1287, 9270, 76225, …
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A121574 → θ+1
1 · 2, 2 · 4, 10, 3 · 8, 32, 24, 4 … row sums 1, 4, 17, 68, 264, 1002, 3739, 13768, …
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A075497 → θ+1
1 · 2, 2 · 4, 12, 3 · 8, 56, 36, 4 … row sums 1, 4, 19, 104, 641, 4380, 32803, …
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A122431 → (θ+1)²
1, 12, 27, 16 · 0, 4, 27, 48, 25 … row sums 56, 104, 168, 248, 344, 456, 584, 728, …
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A108838 → θ+1
2 · 3, 4 · 4, 16, 6 · 5, 40, 45, 8 … row sums 2, 7, 26, 98, 372, 1419, 5434, 20878, …
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A007318 → ×(1+2n+n²) → θ+1
1 · 4, 8 · 9, 36, 27 · 16, 96, 144, 64 … row sums 1, 12, 72, 320, 1200, 4032, 12544, …
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A099097 → (θ+1)²
1 · 0, 12 · 0, 4, 81 · 0, 0, 54, 432 … row sums 1, 12, 85, 486, 2466, 11592, 51667, …
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A143495 → θ+1
1 · 3, 2 · 9, 14, 3 · 27, 74, 36, 4 … row sums 1, 5, 26, 141, 799, 4736, 29371, …
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narayana → (θ+1)²
1 · 1, 4 · 1, 12, 9 · 1, 24, 54, 16 … row sums 1, 5, 22, 95, 406, 1722, 7260, 30459, …
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A099097 → θ+1
1 · 0, 6 · 0, 2, 27 · 0, 0, 18, 108 … row sums 1, 6, 29, 126, 516, 2034, 7807, 29382, …
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A081580 → θ+1
1 · 1, 2 · 1, 14, 3 · 1, 26, 39, 4 … row sums 1, 3, 18, 70, 303, 1197, 4756, 18396, …
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A064861 → θ+1
1, 6, 6 · 1, 10, 24, 16 · 1, 12, 39, 48, 20 … row sums 13, 51, 120, 432, 972, 3348, 7344, …
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A232535 → (θ+1)²
1 · 1, 8 · 1, 32, 27 · 1, 72, 225, 64 … row sums 1, 9, 60, 362, 2032, 10848, 55808, …
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A091044 → ⊙ C(n+2,k+1) → θ+1
1 · 1, 2 · 1, 10, 3 · 1, 28, 42, 4 … row sums 1, 3, 14, 75, 438, 2702, 17324, 114291, …
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A007318 → θ+1 → (θ+1)²
1 · 1, 8 · 1, 16, 27 · 1, 24, 81, 64 … row sums 1, 9, 44, 170, 576, 1792, 5248, 14688, …
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A008292 → γ (scale 1/4) → (θ+1)²
1 · 2 · 4, 8 · 8, 64 … row sums 1, 2, 12, 72, 512, 4144, 37632, 379008, …
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narayana → θ+1
1 · 1, 2 · 1, 6, 3 · 1, 12, 18, 4 … row sums 1, 3, 10, 35, 126, 462, 1716, 6435, …
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A054142 → θ+1
1 · 1, 2 · 1, 6, 3 · 1, 10, 18, 4 … row sums 1, 3, 10, 33, 105, 324, 977, 2895, …
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A066325: Hermite he
1 · 0, 1 · -1, 0, 1 · 0, -3, 0, 1 … row sums 1, 1, 0, -2, -2, 6, 16, -20, …
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basis-map coefficients of A030528, shifted → (θ+1)²
1 · 0, -4 · 0, 0, 9 · 0, 0, 0, -16 … row sums 1, -4, 9, -16, 25, -36, 49, -64, …
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A099040 → θ+1
1 · 0, 4 · 0, 4, 12 · 0, 0, 24, 32 … row sums 1, 4, 16, 56, 188, 608, 1920, 5952, …
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A152198 → θ+1
1 · 1 · 1, 2 · 1, 2 … row sums 1, 1, 3, 3, 8, 8, 20, 20, …
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A086645 → θ+1
1 · 1, 2 · 1, 12, 3 · 1, 30, 45, 4 … row sums 1, 3, 16, 80, 384, 1792, 8192, 36864, …
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A091042 → θ+1
1 · 1, 6 · 1, 20, 15 · 1, 42, 105, 28 … row sums 1, 7, 36, 176, 832, 3840, 17408, 77824, …
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A136572 → θ+1
1 · 0, 2 · 0, 0, 6 · 0, 0, 0, 24 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
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A023531 → D → (θ+1)²
1 · 0, 8 · 0, 0, 27 · 0, 0, 0, 64 … row sums 0, 1, 8, 27, 64, 125, 216, 343, …
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A023531 → 2D+1+2n → θ+1
1 · 2, 6 · 0, 8, 15 · 0, 0, 18, 28 … row sums 1, 8, 23, 46, 77, 116, 163, 218, …
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A100861 → reverse → θ+1
1 · 0, 2 · 0, 2, 3 · 0, 0, 9, 4 … row sums 1, 2, 5, 13, 38, 116, 382, 1310, …
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A008292 → γ (scale 1/2) (shift +1) → θ+1
1 · 1 · 1, 2 · 1, 8 … row sums 1, 1, 3, 9, 35, 155, 791, 4529, …
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A082985 → (θ+1)²
1 · 1, 12 · 1, 20, 45 · 1, 28, 126, 112 … row sums 1, 13, 66, 267, 985, 3444, 11597, …
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A007318 → θ+1+n → (θ+1)²
1 · 2, 12 · 3, 32, 45 · 4, 60, 162, 112 … row sums 1, 14, 80, 338, 1216, 3952, 11968, …
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A062264 → (θ+1)²
1 · 1, 20 · 1, 48, 135 · 1, 84, 567, 560 … row sums 1, 21, 184, 1212, 6975, 37147, 188188, …