preserver_image / reversal, 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
- hadamard_multiplier, shift -1 2 proof steps, 2 sequences
- affine, shift 2 2 proof steps, 2 sequences
- euler_polynomial, 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
164 sequences
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A062264 → reverse
1 · 5, 1 · 15, 12, 1 · 35, 63, 21, 1 … row sums 1, 6, 28, 120, 495, 2002, 8008, 31824, …
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A130534: Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, …
1 · 1, 1 · 2, 3, 1 · 6, 11, 6, 1 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
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A227450 → reverse
2 · 4, 2 · 8, 12, 2 · 16, 48, 28, 2 … row sums 2, 6, 22, 94, 454, 2430, 14214, 89918, …
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A039755 → reverse
1 · 1, 1 · 1, 4, 1 · 1, 9, 13, 1 … row sums 1, 2, 6, 24, 116, 648, 4088, 28640, …
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A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
A007318 → D+11 · 2, 1 · 3, 4, 1 · 4, 9, 6, 1 … row sums 1, 3, 8, 20, 48, 112, 256, 576, …
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A081577: Pascal-(1,2,1) array read by antidiagonals.
1 · 1, 1 · 1, 4, 1 · 1, 7, 7, 1 … row sums 1, 2, 6, 16, 44, 120, 328, 896, …
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basis-map coefficients of A026729 → basis A056242 → reverse
1 · 2, 1 · 4, 5, 1 · 8, 16, 9, 1 … row sums 1, 3, 10, 34, 116, 396, 1352, 4616, …
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A008278: Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
1 · 1, 1 · 1, 3, 1 · 1, 6, 7, 1 … row sums 1, 2, 5, 15, 52, 203, 877, 4140, …
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A122848: Exponential Riordan array (1, x(1+x/2)).
A100861 → reverse1 · 0, 1 · 0, 1, 1 · 0, 0, 3, 1 … row sums 1, 1, 2, 4, 10, 26, 76, 232, …
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A086645: Triangle read by rows: T(n, k) = binomial(2n, 2k).
1 · 1, 1 · 1, 6, 1 · 1, 15, 15, 1 … row sums 1, 2, 8, 32, 128, 512, 2048, 8192, …
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A082985 → reverse
1 · 3, 1 · 5, 5, 1 · 7, 14, 7, 1 … row sums 1, 4, 11, 29, 76, 199, 521, 1364, …
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A103327 → reverse
1 · 1, 3 · 1, 10, 5 · 1, 21, 35, 7 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
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A008292: Eulerian
1 · 1, 1 · 1, 4, 1 · 1, 11, 11, 1 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
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A091042 → reverse
1 · 3, 1 · 5, 10, 1 · 7, 35, 21, 1 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …