Real-rooted atlas

A059364 → basis A007318 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1437
228451891
32807025851621729
4364012486160299126194443225
558240254376443754386451167994291601339975
611065605881464130100941533014110148004357793252488049579075
72434432015262996840973281261047507654518904929182258886683932110224802131900225

All 15 rows

Technical details
Node id
n-7345054e428f3912 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-13e69f2906ee7e95

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:           4,            3;
  2:          28,           45,           18;
  3:         280,          702,          585,           162;
  4:        3640,        12486,        16029,          9126,          1944;
  5:       58240,       254376,       443754,        386451,        167994,         29160;
  6:     1106560,      5881464,     13010094,      15330141,      10148004,       3577932,       524880;
  7:    24344320,    152629968,    409732812,     610475076,     545189049,     291822588,     86683932,     11022480;
  8:   608608000,   4400012880,  13906439532,   25095464388,   28281128049,   20380101876,   9170840412,   2355976368,   264539520;
  9: 17041024000, 139632776640, 508180654656, 1078146870228, 1469449123848, 1334233309851, 807046282188, 313580029428, 71018468496, 7142567040;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+1+3*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[(3*n + 1 + 3*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] = (3*n + 1 + 3*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

Same fingerprint ?

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.