Proof step e-337faa044bd16e1a

preserver_image derivative_of_power_multiple, shift 1 → concludes about Triangle related to the asymptotic expansion of E(x,m=3,n). JSON

Theorem

Rolle’s theorem (elementary) polytool ?

Lean (RealRooted): RealRooted.DegreeDropReversal.splits_X_pow_mul_iff, Polynomial.Splits.iterate_derivative

Operation

Image under a real-rootedness preserver

{
  "index_shift": 1,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"derivative_of_power_multiple\",\"power\":1,\"derivative_order\":2}"
}

Premises

• 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… (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "0" } } }
• { "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 2e328668c98fc5fa271037fd38c6fffaad254d8a0fdd48e4d0d75d8d6f083ad4, 12,253 bytes (3,145 compressed); download JSON

Cited by

• n-2c8bb380cc75972b (real_rooted)
• n-2c8bb380cc75972b (one_signed_coefficients)
• n-2c8bb380cc75972b (hadamard_kernel)
• n-2c8bb380cc75972b (interlacing)