Proof step
Pólya–Schur theorem on multiplier sequences
Hadamard product with a multiplier sequence binom(n+1, k+1)
- Concludes
- Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
- 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-748356502f20a413· JSON- Operation
- Hadamard product with a multiplier sequence
{ "factor": "binom(n+1, k+1)", "operation": "hadamard_factor" } - Claims
{ "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "0" } } }- Certificate
hadamard_factor, SHA-256bcb506e8c4da33841805f5ca07363d408fe2653f2305558d14ab22503b91ce90, 9,340 bytes (2,246 compressed); downloadpolytool replayed the certificate when the step was written.- Cited by
n-511a2225fa2191c9(real_rooted)n-511a2225fa2191c9(one_signed_coefficients)n-511a2225fa2191c9(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.