Proof step
Pólya–Schur theorem on multiplier sequences
Hadamard product with a multiplier sequence binom(n+2, k)
- Concludes
- Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
- Premises
- a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
- a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
- Theorem
- [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ?
- Checked
- polytool replayed · Lean not attempted
Technical details
- Step id
e-c0ec1f8037a12781· JSON- Operation
- Hadamard product with a multiplier sequence
{ "factor": "binom(n+2, k)", "operation": "hadamard_factor" } - Claims
{ "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "0" } } }- Certificate
hadamard_factor, SHA-256aac422244982c34923e18e1ce0ce409ce56fca0f42bf081b5011f740a4fb3f8d, 7,653 bytes (2,084 compressed); downloadpolytool replayed the certificate when the step was written.- Cited by
n-d51e4f59f14ff00f(real_rooted)n-d51e4f59f14ff00f(one_signed_coefficients)n-d51e4f59f14ff00f(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.