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_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
- 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
352 sequences
-
A034870 → θ+1
1 · 1, 4, 3 · 1, 8, 18, 16, 5 … row sums 1, 8, 48, 256, 1280, 6144, 28672, …
-
A100861 → (θ+1)²
1 · 1 · 1, 4 · 1, 12 … row sums 1, 1, 5, 13, 52, 176, 706, 2710, …
-
A130534 → θ+1
1 · 1, 2 · 2, 6, 3 · 6, 22, 18, 4 … row sums 1, 3, 11, 50, 274, 1764, 13068, 109584, …
-
A113214 → θ+1
1 · 2, 2 · 0, 6, 3 · 0, 4, 12, 4 … row sums 1, 4, 9, 20, 40, 78, 147, 272, …
-
A023531 → (1/2)D²+2D+1 → θ+1
1 · 2, 2 · 1, 8, 3 · 0, 6, 18, 4 … row sums 1, 4, 12, 28, 55, 96, 154, 232, …
-
A034870 → (θ+1)²
1 · 1, 8, 9 · 1, 16, 54, 64, 25 … row sums 1, 18, 160, 1120, 6912, 39424, 212992, …
-
A106800 → θ+1
1 · 1 · 1, 2 · 1, 6, 3 … row sums 1, 1, 3, 10, 38, 161, 747, 3753, …
-
A001263 → (θ+1)²
1 · 1, 4 · 1, 12, 9 · 1, 24, 54, 16 … row sums 1, 5, 22, 95, 406, 1722, 7260, 30459, …
-
A274496 → (θ+1)²
1 · 2 · 2, 8 · 4, 16 … row sums 1, 2, 10, 20, 72, 144, 448, 896, …
-
A027465 → θ+1
1 · 3, 2 · 9, 12, 3 · 27, 54, 27, 4 … row sums 1, 5, 24, 112, 512, 2304, 10240, 45056, …
-
A094816 → (θ+1)²
1 · 1, 4 · 1, 12, 9 · 1, 32, 54, 16 … row sums 1, 5, 22, 103, 543, 3273, 22512, …
-
A152842 → θ+1
1 · 1, 6 · 1, 8, 9 · 1, 14, 45, 36 … row sums 1, 7, 18, 96, 224, 1088, 2432, 11264, …
-
A094638 → (θ+1)²
1 · 1, 4 · 1, 12, 18 · 1, 24, 99, 96 … row sums 1, 5, 31, 220, 1756, 15596, 152804, …
-
A008277 → θ+2 (shift +1) → θ+1
2 · 2, 6 · 2, 18, 12 · 2, 42, 72, 20 … row sums 2, 8, 32, 136, 622, 3060, 16130, 90688, …
-
A103327 → θ+1
1 · 3, 2 · 5, 20, 3 · 7, 70, 63, 4 … row sums 1, 5, 28, 144, 704, 3328, 15360, 69632, …
-
A121574 → (θ+1)²
1 · 2, 4 · 4, 20, 9 · 8, 64, 72, 16 … row sums 1, 6, 33, 160, 726, 3144, 13159, 53652, …
-
A091044 → (θ+1)²
1 · 2, 8 · 3, 40, 27 · 4, 112, 252, 64 … row sums 1, 10, 70, 432, 2464, 13312, 69120, …
-
A169803 → θ+1
1 · 1, 2 · 1, 4 · 1, 6, 3 … row sums 1, 3, 5, 10, 18, 33, 59, 105, …
-
A116414 → θ+1
1 · 4, 2 · 13, 16, 3 · 40, 84, 36, 4 … row sums 1, 6, 32, 164, 819, 4012, 19363, 92352, …
-
A007318 → ×(1+2n+n²) → (θ+1)²
1 · 4, 16 · 9, 72, 81 · 16, 192, 432, 256 … row sums 1, 20, 162, 896, 4000, 15552, 54880, …
-
A120910 → θ+1
3 · 6, 6 · 12, 24, 9 · 24, 72, 54, 12 … row sums 3, 12, 45, 162, 567, 1944, 6561, 21870, …
-
A207823 → θ+1
1 · 4, 2 · 15, 16, 3 · 56, 92, 36, 4 … row sums 1, 6, 34, 188, 1021, 5468, 28959, …
-
A008292 → θ+1
1 · 1, 2 · 1, 8, 3 · 1, 22, 33, 4 … row sums 0, 1, 3, 12, 60, 360, 2520, 20160, …
-
A052553 → θ+1
1 · 1 · 1, 2 · 1, 4 … row sums 1, 1, 3, 5, 10, 18, 33, 59, …
-
A266213 → (θ+1)²
1 · 1 · 1, 8 · 1, 16, 18 … row sums 1, 1, 9, 35, 129, 437, 1409, 4375, …
-
A010054 → reverse → θ+1
1 · 0, 8 · 0, 0, 21 · 0, 0, 0, 40 … row sums 1, 8, 21, 40, 65, 96, 133, 176, …
-
A010054 → reverse → (θ+1)²
1 · 0, 16 · 0, 0, 63 · 0, 0, 0, 160 … row sums 1, 16, 63, 160, 325, 576, 931, 1408, …
-
A188440 → θ+1
1 · 1 · 1, 4 · 1, 4 … row sums 1, 1, 5, 5, 21, 21, 81, 81, …
-
A249251 → θ+1
1 · 1, 6 · 1, 12, 18 · 1, 18, 63, 48 … row sums 1, 7, 31, 130, 520, 2017, 7651, 28537, …
-
A007318 → (θ+1)² → θ+1
1 · 1, 8 · 1, 16, 27 · 1, 24, 81, 64 … row sums 1, 9, 44, 170, 576, 1792, 5248, 14688, …
-
A145596 → θ+1
1 · 2, 4 · 3, 16, 9 · 4, 40, 60, 16 … row sums 1, 6, 28, 120, 495, 2002, 8008, 31824, …
-
A126351 → θ+1
1 · 1, 4 · 1, 10, 12 · 1, 18, 57, 32 … row sums 1, 5, 23, 108, 534, 2803, 15623, 92249, …
-
basis-map coefficients of A026729 → basis A056242 → (θ+1)²
1 · 1, 8 · 1, 20, 36 · 1, 36, 144, 128 … row sums 1, 9, 57, 309, 1530, 7134, 31864, …
-
A120909 → θ+1
3 · 3, 12 · 3, 24, 36 · 3, 36, 108, 96 … row sums 3, 15, 63, 243, 891, 3159, 10935, …
-
A011973 → basis A007318 → θ+1
1 · 1 · 2, 2 · 3, 4 … row sums 1, 1, 4, 7, 18, 37, 84, 179, …
-
A099093 → θ+1
1 · 0, 6 · 0, 6, 27 · 0, 0, 54, 108 … row sums 1, 6, 33, 162, 756, 3402, 14931, 64314, …
-
A274498 → θ+1
1 · 3 · 3, 12 · 9, 36 … row sums 1, 3, 15, 45, 189, 567, 2187, 6561, …
-
A013610 → θ+1
1 · 1, 6 · 1, 12, 27 · 1, 18, 81, 108 … row sums 1, 7, 40, 208, 1024, 4864, 22528, …
-
A249248 → θ+1
1 · 1, 6 · 1, 14, 9 · 1, 24, 66, 12 … row sums 1, 7, 24, 103, 419, 1815, 7970, 36277, …
-
A125662 → (θ+1)²
1 · 3, 4 · 8, 24, 9 · 21, 100, 81, 16 … row sums 1, 7, 41, 218, 1091, 5237, 24379, …
-
A013620 → (θ+1)²
1 · 2, 12 · 4, 48, 81 · 8, 144, 486, 432 … row sums 1, 14, 133, 1070, 7825, 53750, 353125, …
-
A116666: Triangle, row sums = number of edges in n-dimensional hypercubes.
A007318 → 2θ+11 · 1, 3 · 1, 6, 5 · 1, 9, 15, 7 … row sums 1, 4, 12, 32, 80, 192, 448, 1024, …
-
A122542 → (θ+1)²
1 · 0, 4 · 0, 8, 9 · 0, 8, 36, 16 … row sums 1, 4, 17, 60, 201, 640, 1969, 5896, …
-
A165253 → (θ+1)²
1 · 1 · 1, 4 · 1, 12, 9 … row sums 1, 1, 5, 22, 86, 313, 1085, 3629, …
-
A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
A010054 → D∘x1 · 3 · 5 · 7 … row sums 1, 3, 5, 7, 9, 11, 13, 15, …
-
A143494 → θ+1
1 · 2, 2 · 4, 10, 3 · 8, 38, 27, 4 … row sums 1, 4, 17, 77, 372, 1915, 10481, 60814, …
-
A116412 → θ+1
1 · 3, 2 · 6, 12, 3 · 12, 42, 27, 4 … row sums 1, 5, 21, 85, 332, 1266, 4741, 17507, …
-
A172431 → ⊙ C(n,k+1) (shift -1) → (θ+1)²
1 · 1, 4 · 1, 16, 9 · 1, 36, 90, 16 … row sums 1, 5, 26, 143, 788, 4332, 23735, …
-
A344565 → θ+1
3 · 15, 30 · 45, 180, 135 · 105, 630, 945, 420 … row sums 3, 45, 360, 2100, 10080, 42336, 161280, …
-
A059344 → ⊙ 1/k! (shift -1) → θ+1
1 · 1 · 1, 4 · 1, 12 … row sums 1, 1, 5, 13, 43, 131, 411, 1275, …