Proof step e-422298c8a4851a11

basis_map A008288 → concludes about basis_map_A008288 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows. JSON

Theorem

Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A008288",
  "operation": "basis_map"
}

Premises

• T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows. (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "lower": { "closed": true, "exact": { "radicand": "2", "rational_part": "-3", "sqrt_coefficient": "-2" }, "value": "-1564556493/268435456" }, "upper": { "closed": true, "exact": { "radicand": "2", "rational_part": "-3", "sqrt_coefficient": "2" }, "value": "-46056243/268435456" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 8ec9791e800f857d1b6c0908f4a4be1d73fa83371ed0e3be628ab3509a62f7e6, 24,234 bytes (3,200 compressed); download JSON

Cited by

• n-59aff779bb582ee4 (one_signed_coefficients)
• n-59aff779bb582ee4 (hadamard_kernel)
• n-59aff779bb582ee4 (real_rooted)