Real-rooted atlas

A088996 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-1 + 6\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
151217
255252288595
39356264139681036831535
4215051889526219369089284976642238985
5623645676990829373264636750726896793629859840199269665
6218275752818492201515497616434350252869984829446010785792214990848021321854155
789493057513389334320858107367363053854033926517920130568343147847685930522542081805923123202665231769375

All 15 rows

Technical details
Node id
n-447b8fa6e8cf1eaa · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-47aff865bb54210c

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:             5,             12;
  2:            55,            252,             288;
  3:           935,           6264,           13968,            10368;
  4:         21505,         188952,          621936,           908928,           497664;
  5:        623645,        6769908,        29373264,         63675072,         68967936,          29859840;
  6:      21827575,      281849220,      1515497616,       4343502528,       6998482944,        6010785792,        2149908480;
  7:     894930575,    13389334320,     85810736736,     305385403392,     651792013056,      834314784768,      593052254208,      180592312320;
  8:   42061737025,   715212048240,   5318480721312,   22590944686080,   59951223339264,   101784828137472,   107967675285504,    65420855083008,   17336861982720;
  9: 2229272062325, 42448906155420, 359122379439456, 1771715986263936, 5617236863077632, 11869328011926528, 16715048228978688, 15127814250233856, 7984306034049024, 1872381094133760;
%F Formula
The row polynomials satisfy P(n,x) = (6*n-1+12*n*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[(6*n - 1 + 12*n*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] = (6*n - 1 + 12*n*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.