Real-rooted atlas

A038328 → 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

\(P_{n}(x) = \left(16 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 64\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0256644324
14096153619285832
26553632768614451216104976
31048576655360163840204801280321889568
4167772161258291239321606553606144030726434012224
5268435456234881024880803841835008022937601720327168128612220032
64294967296429496729618790481924697620487340032073400324587521638425611019960576
768719476736773094113283865470566411274289152211392921626424115222020096117964836864512198359290368

All 15 rows

Technical details
Node id
n-d89fc0d0973b1556 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-8ca1e631330e3f91

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:            256,             64,              4;
  1:           4096,           1536,            192,             8;
  2:          65536,          32768,           6144,           512,            16;
  3:        1048576,         655360,         163840,         20480,          1280,           32;
  4:       16777216,       12582912,        3932160,        655360,         61440,         3072,          64;
  5:      268435456,      234881024,       88080384,      18350080,       2293760,       172032,        7168,        128;
  6:     4294967296,     4294967296,     1879048192,     469762048,      73400320,      7340032,      458752,      16384,       256;
  7:    68719476736,    77309411328,    38654705664,   11274289152,    2113929216,    264241152,    22020096,    1179648,     36864,     512;
  8:  1099511627776,  1374389534720,   773094113280,  257698037760,   56371445760,   8455716864,   880803840,   62914560,   2949120,   81920,   1024;
  9: 17592186044416, 24189255811072, 15118284881920, 5669356830720, 1417339207680, 248034361344, 31004295168, 2768240640, 173015040, 7208960, 180224, 2048;
%F Formula
The row polynomials satisfy P(n,x) = (16+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 64*x + 4*x^2.
%t Mathematica
P[0, x_] := 256 + 64*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 2*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] = 256 + 64*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 2*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.