Proof step
Pólya–Schur theorem on multiplier sequences
Hadamard product with a multiplier sequence binom(n+1, k)
- Concludes
- Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).
- Premises
- hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).
sequence: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
- hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).
- Theorem
- [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ?
- Checked
- polytool replayed · Lean not attempted
Technical details
- Step id
e-883bba4ec6cabdf4· JSON- Operation
- Hadamard product with a multiplier sequence
{ "factor": "binom(n+1, k)", "operation": "hadamard_factor" } - Claims
{ "claim": "real_rooted", "from_index": 1, "region": { "upper": { "closed": true, "value": "0" } } }- Certificate
hadamard_factor, SHA-256edc612ac8ff3f82730f01281e9119efe43c6c8a5e46d54e9d7d8f75830775a2d, 20,652 bytes (5,160 compressed); downloadpolytool replayed the certificate when the step was written.- Cited by
n-b002679da3d71a0f(real_rooted)n-b002679da3d71a0f(one_signed_coefficients)n-b002679da3d71a0f(hadamard_kernel)
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.