Proof step e-bb56aa5752b1a607

basis_map A007318 → concludes about shift_x_plus_one of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals. JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A007318",
  "operation": "basis_map"
}

Premises

• Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals. (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": 1, "region": { "upper": { "closed": true, "value": "0" } } }
• { "claim": "interlacing", "from_index": 2, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 63e0e8f48e37ed463d5bab104db175d948ba4601c8f5acd99135a0b0feb2b588, 10,718 bytes (2,412 compressed); download JSON

Cited by

• n-622af52a4165c278 (hadamard_kernel)
• n-622af52a4165c278 (real_rooted)
• n-622af52a4165c278 (interlacing)
• n-622af52a4165c278 (one_signed_coefficients)