Real-rooted atlas

A137320 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(5 + 4\,x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
2425216110
3336584336641320
4302466005360192025617160
530240780968000040640102401024240240
633264098001611923847670402752005222440963603600
73991680130907521822867213974016637056017274882580481638457657600

All 15 rows

Technical details
Node id
n-9e1c03cafdcaad35 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-c714186966d687c0

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         6,          4;
  2:        42,         52,         16;
  3:       336,        584,        336,         64;
  4:      3024,       6600,       5360,       1920,        256;
  5:     30240,      78096,      80000,      40640,      10240,       1024;
  6:    332640,     980016,    1192384,     767040,     275200,      52224,      4096;
  7:   3991680,   13090752,   18228672,   13974016,    6370560,    1727488,    258048,    16384;
  8:  51891840,  186146496,  289335744,  254576896,  138713344,   47939584,  10264576,  1245184,   65536;
  9: 726485760, 2813618304, 4795286400, 4721419520, 2960294400, 1226007552, 335462400, 58490880, 5898240, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (n+5+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + 5 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (n + 5 + 4*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.