Proof step
Pólya–Schur theorem on multiplier sequences
Hadamard product with a multiplier sequence 1/k!
- Concludes
- Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).
- Premises
- Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
sequence: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
- Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
- Theorem
- [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ?
- Checked
- polytool replayed · Lean not attempted
Technical details
- Step id
e-e3b364f94f508324· JSON- Operation
- Hadamard product with a multiplier sequence
{ "factor": "1/k!", "operation": "hadamard_factor" } - Claims
{ "claim": "real_rooted", "from_index": 1, "region": { "upper": { "closed": true, "value": "0" } } }- Certificate
hadamard_factor, SHA-256be95ee08374957235d078ec3104ccf9c2cdaffb31dee941ec1cc87a4554b5647, 6,471 bytes (1,832 compressed); downloadpolytool replayed the certificate when the step was written.- Cited by
n-f25b32370a8d627e(real_rooted)n-f25b32370a8d627e(one_signed_coefficients)n-f25b32370a8d627e(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.