Real-rooted atlas

A038329 → x ↦ x+1 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 324 + 108\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
03241089441
158322916486279261
21049766998417496194481194481
3188956815746405248808748072902434084101
434012224340122241417176031492803936602624472985766121
5612220032714256704357128352992023201653372016533729185421871801088541
6110199605761469328076885710804482857026816595213920793618566613488314928656137822859361
71983592903682975389355521983592903687713972403219284931008321415516835712835225509168106288219683794280046581

All 15 rows

Technical details
Node id
n-95ff44822e25b07f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a5035a1c3b326aa3

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:            324,             108,              9;
  1:           5832,            2916,            486,             27;
  2:         104976,           69984,          17496,           1944,             81;
  3:        1889568,         1574640,         524880,          87480,           7290,           243;
  4:       34012224,        34012224,       14171760,        3149280,         393660,         26244,          729;
  5:      612220032,       714256704,      357128352,       99202320,       16533720,       1653372,        91854,        2187;
  6:    11019960576,     14693280768,     8571080448,     2857026816,      595213920,      79361856,      6613488,      314928,       6561;
  7:   198359290368,    297538935552,   198359290368,    77139724032,    19284931008,    3214155168,    357128352,    25509168,    1062882,     19683;
  8:  3570467226624,   5950778711040,  4463084033280,  1983592903680,   578547930240,  115709586048,  16070775840,  1530550080,   95659380,   3542940,    59049;
  9: 64268410079232, 117825418478592, 98187848732160, 49093924366080, 16364641455360, 3818416339584, 636402723264, 75762228960, 6313519080, 350751060, 11691702, 177147;
%F Formula
The row polynomials satisfy P(n,x) = (18+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 108*x + 9*x^2.
%t Mathematica
P[0, x_] := 324 + 108*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(18 + 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] = 324 + 108*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 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

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.