Real-rooted atlas

A145324 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5 + 3\,x + 4\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
19615
211715954330
31989410728266489570
441769116082120951559989720344520
51044225365389251132513577068125096417496014814360
630282525127891593225016011211113243111396384313440843674160740718000
799932332549472031691049492659512367121283874279850437078679888735052968817984042220926000

All 15 rows

Technical details
Node id
n-9eb1bb222a47d2ea · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-df938e0c3ee60fbd

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:             9,             6;
  2:           117,           159,             54;
  3:          1989,          4107,           2826,            648;
  4:         41769,        116082,         120951,          55998,           9720;
  5:       1044225,       3653892,        5113251,        3577068,        1250964,         174960;
  6:      30282525,     127891593,      225016011,      211113243,      111396384,       31344084,        3674160;
  7:     999323325,    4947203169,    10494926595,    12367121283,     8742798504,     3707867988,      873505296,      88179840;
  8:   36974963025,  210028247028,   521886769578,   740946505536,   657395819289,   373246675164,   132432131628,   26847297072,   2380855680;
  9: 1515973484025, 9720407018898, 27698204963538, 46035409814316, 49181623756929, 35024988260394, 16627117651668, 5073703128792, 903033995040, 71425670400;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+5+(3*n+3)*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[(4*n + 5 + (3*n + 3)*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] = (4*n + 5 + (3*n + 3)*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.