preserver_image / derivative_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
- 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
410 sequences
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A059344 → D+1
1 · 1 · 3, 2 · 7, 6 … row sums 1, 1, 5, 13, 61, 221, 1081, 4705, …
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A066325 → D
1 · 0, 2 · -3, 0, 3 · 0, -12, 0, 4 … row sums 0, 1, 2, 0, -8, -10, 36, 112, …
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A106800 → D+1
1 · 1 · 2, 1 · 4, 5, 1 … row sums 1, 1, 3, 10, 38, 161, 747, 3753, …
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A053123 → D
-2 · -4, 6 · -6, 20, -12 · -8, 42, -60, 20 … row sums 0, -2, 2, 2, -6, 4, 4, -10, …
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A091044 → D
2 · 10, 6 · 28, 56, 12 · 60, 252, 180, 20 … row sums 0, 2, 16, 96, 512, 2560, 12288, 57344, …
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A038231 → D+1
1 · 5, 1 · 24, 10, 1 · 112, 72, 15, 1 … row sums 1, 6, 35, 200, 1125, 6250, 34375, …
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A054142 → D+1
1 · 2, 1 · 4, 5, 1 · 6, 17, 9, 1 … row sums 1, 3, 10, 33, 105, 324, 977, 2895, …
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A125662 → D+1
1 · 4, 1 · 14, 8, 1 · 46, 43, 12, 1 … row sums 1, 5, 23, 102, 441, 1871, 7823, 32332, …
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narayana → γ (shift +1) → D+1
1 · 1 · 2, 1 · 4, 3 … row sums 1, 1, 3, 7, 19, 51, 141, 393, …
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A344565 → D+1
3 · 30, 15 · 135, 180, 45 · 420, 945, 630, 105 … row sums 3, 45, 360, 2100, 10080, 42336, 161280, …
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A172431 → ⊙ C(n,k+1) (shift -1) → D+1
1 · 2, 1 · 5, 6, 1 · 10, 29, 13, 1 … row sums 1, 3, 12, 53, 244, 1152, 5531, 26875, …
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A023531 → (θ+1)² → D
4 · 0, 18 · 0, 0, 48 · 0, 0, 0, 100 … row sums 0, 4, 18, 48, 100, 180, 294, 448, …
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A046716 → D+1
1 · 2, 1 · 4, 5, 1 · 7, 22, 11, 1 … row sums 1, 3, 10, 41, 209, 1287, 9270, 76225, …
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A143495: Triangle read by rows: 3-Stirling numbers of the second kind.
1 · 3, 1 · 9, 7, 1 · 27, 37, 12, 1 … row sums 1, 4, 17, 77, 372, 1915, 10481, 60814, …
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A143685 → D+1
1 · 2, 1 · 12, 13, 1 · 22, 63, 24, 1 … row sums 1, 3, 26, 110, 615, 2821, 13828, 63756, …
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A130534 → D
1 · 3, 2 · 11, 12, 3 · 50, 70, 30, 4 … row sums 0, 1, 5, 26, 154, 1044, 8028, 69264, …
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A007318 → −D²+(−1+n)D+1+n → D
2 · 8, 6 · 18, 36, 12 · 32, 108, 96, 20 … row sums 0, 2, 14, 66, 256, 880, 2784, 8288, …
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A010054 → reverse → D
2 · 0, 6 · 0, 0, 12 · 0, 0, 0, 20 … row sums 0, 0, 2, 6, 12, 20, 30, 42, …
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A102547 → reverse → D+1
1, 1 · 0, 2, 1 · 0, 0, 3, 1 · 0, 0, 3, 5, 1 … row sums 2, 3, 4, 9, 16, 25, 42, 70, …
-
A111125 → D
1 · 5, 2 · 14, 14, 3 · 30, 54, 27, 4 … row sums 0, 1, 7, 31, 115, 390, 1254, 3893, …
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A007318 → −D²+(−1+n)D+1+n → D+1
1 · 4, 2 · 11, 14, 3 · 22, 54, 30, 4 … row sums 1, 6, 28, 110, 384, 1232, 3712, 10656, …
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A081581 → D+1
1 · 2, 1 · 9, 10, 1 · 16, 45, 18, 1 … row sums 1, 3, 20, 80, 372, 1540, 6496, 26496, …
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A249251 → D
3 · 6, 12 · 9, 42, 36 · 12, 90, 180, 96 … row sums 0, 3, 18, 87, 378, 1548, 6102, 23421, …
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A001263 → D
1 · 3, 2 · 6, 12, 3 · 10, 40, 30, 4 … row sums 0, 1, 5, 21, 84, 330, 1287, 5005, …
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A082985 → D+1
1 · 4, 3 · 6, 15, 5 · 8, 35, 35, 7 … row sums 1, 7, 26, 85, 265, 804, 2393, 7019, …
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A035607 → D+1
1 · 1 · 3, 2 · 5, 8, 2 … row sums 1, 1, 5, 15, 45, 129, 361, 991, …
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A122076 → D
2 · 8, 4 · 30, 30, 6 · 104, 160, 72, 8 … row sums 0, 2, 12, 66, 344, 1710, 8196, 38234, …
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A108838 → D
2 · 8, 4 · 20, 30, 6 · 40, 120, 72, 8 … row sums 0, 2, 12, 56, 240, 990, 4004, 16016, …
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basis-map coefficients of A113214, shifted → D+1
1 · -2, -1 · 3, 4, 1 · -4, -9, -6, -1 … row sums 1, -3, 8, -20, 48, -112, 256, -576, …
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type_b_eulerian → γ (scale 1/4) → D
1 · 5 · 18, 10 · 58, 122 … row sums 0, 0, 1, 5, 28, 180, 1320, 10920, …
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A008292 → γ (scale 1/2) (shift +1) → D+1
1 · 1 · 2, 1 · 5, 4 … row sums 1, 1, 3, 9, 35, 155, 791, 4529, …
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A132013: T(n,j) for an iterated mixed order Laguerre transform. Coefficients of the normalized generalized Laguerre polynomials (-1)^n*n!*L(n,1-n,x).
A023531 → (−1/n)D²+(1/n)D1 · -1, 1 · 0, -2, 1 · 0, 0, -3, 1 … row sums 1, 0, -1, -2, -3, -4, -5, -6, …
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A120910 → D
3 · 12, 6 · 36, 36, 9 · 96, 144, 72, 12 … row sums 0, 3, 18, 81, 324, 1215, 4374, 15309, …
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basis-map coefficients of A026729 → basis A056242 → D
2 · 5, 8 · 9, 32, 24 · 14, 82, 132, 64 … row sums 0, 2, 13, 65, 292, 1236, 5036, 19980, …
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A108299 → D+1
1 · 0, -1 · 0, -3, -1 · 0, -5, 1, 1 … row sums 1, -1, -4, -3, 3, 8, 5, -5, …
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A064861 → D+1
4, 7, 2 · 6, 21, 20, 4 · 7, 32, 49, 28, 4 … row sums 13, 51, 120, 432, 972, 3348, 7344, …
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A008459 → D
1 · 4, 2 · 9, 18, 3 · 16, 72, 48, 4 … row sums 0, 1, 6, 30, 140, 630, 2772, 12012, …
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A008292 → γ (scale 1/2) (shift +1) → D
1 · 4 · 11, 8 · 26, 68 … row sums 0, 0, 1, 4, 19, 94, 519, 3144, …
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basis-map coefficients of A026729 → D+1
1 · 1, 1 · 0, 2, 1 · 0, 0, 3, 1 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
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A062145 → D
4 · 10, 20 · 18, 90, 60 · 28, 252, 420, 140 … row sums 0, 4, 30, 168, 840, 3960, 18018, 80080, …
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A007318 → 2θ+1 → D+1
1 · 4, 3 · 7, 16, 5 · 10, 39, 36, 7 … row sums 1, 7, 28, 92, 272, 752, 1984, 5056, …
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A207823 → D+1
1 · 5, 1 · 23, 10, 1 · 102, 70, 15, 1 … row sums 1, 6, 34, 188, 1021, 5468, 28959, …
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A188440 → D
2 · 2 · 4, 8 · 4, 8 … row sums 0, 0, 2, 2, 12, 12, 54, 54, …
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A007318 → basis A049020 → D
1 · 5, 2 · 22, 18, 3 · 99, 122, 42, 4 … row sums 0, 1, 7, 43, 267, 1731, 11819, 85107, …
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A272866 → D
3, 2 · 6, 22, 18, 4 · 9, 60, 135, 120, 45, 6 … row sums 0, 5, 50, 375, 2500, 15625, 93750, …
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A100861 → reverse → D
1 · 1, 2 · 0, 6, 3 · 0, 6, 18, 4 … row sums 0, 1, 3, 9, 28, 90, 306, 1078, …
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A007318 → (1/2+(1/2)n)θ+1+n → D
3 · 9, 12 · 18, 48, 30 · 30, 120, 150, 60 … row sums 0, 3, 21, 96, 360, 1200, 3696, 10752, …
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A008277 → θ+2 (shift +1) → D+1
2 · 5, 3 · 11, 17, 4 · 23, 69, 39, 5 … row sums 2, 8, 32, 136, 622, 3060, 16130, 90688, …
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A391533 → D
4, 2 · 8, 10, 3 · 12, 26, 18, 4 · 16, 50, 57, 28, 5 … row sums 6, 21, 60, 156, 384, 912, 2112, 4800, …
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A144438 → D+1
1 · 2, 1 · 6, 7, 1 · 15, 42, 17, 1 … row sums 1, 3, 14, 75, 471, 3402, 27844, 254970, …