preserver_image / affine, shift -1
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 -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
- derivative_of_power_multiple, shift 0 1 proof steps, 1 sequences
- derivative_of_power_multiple, shift 1 1 proof steps, 1 sequences
5 sequences
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A039757: Triangle of coefficients in expansion of (x-1)*(x-3)*(x-5)*...*(x-(2*n-1)).
A028338 → x ↦ −x (shift -1)1 · -1, 1 · 3, -4, 1 · -15, 23, -9, 1 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
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A021010: Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).
A144084 → x ↦ −x (shift -1)1 · -1, 1 · 1, -4, 2 · -1, 9, -18, 6 … row sums 1, 0, -1, -4, -15, -56, -185, -204, …
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A048994: Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0 <= k <= n.
A028338 → x ↦ −2x−1 (shift -1)1 · 0, 1 · 0, -1, 1 · 0, 2, -3, 1 … row sums 1, 1, 0, 0, 0, 0, 0, 0, …
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A008276: Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n)/x in powers of x.
A059364 → x ↦ −x−1 (shift -1)1 · 1, -1 · 1, -3, 2 · 1, -6, 11, -6 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
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A159834: Coefficient array of Hermite_H(n, (x-1)/sqrt(2))/(sqrt(2))^n.
A066325 → x ↦ x−1 (shift -1)1 · -1, 1 · 0, -2, 1 · 2, 0, -3, 1 … row sums 1, 0, -1, 0, 3, 0, -15, 0, …