Real-rooted atlas

A159854 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 5\,x + 3\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1 + 4\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01449
1211201245
2428738436225
38682303873241081125
4161606641464180911883245625
5323681808492080107803421297228125
664832472015264307803963631833145802916140625
712818561193644688107352171612182574124659495728748703125
8256409629440125184348768665280879984797040473121166212262443515625
95128960711683386881073088237686437558084234032333736217517875511247873217578125
1010241945616921689088031622407972992146422081973548819376820135156606357609181083623619687890625
1120484198439680022894088997120254327045320339283397600979601048516178053262198226944995904900708588439453125
1240969011291955257692162486246477856768182704896326405376446113008464203872362009736205175592798932971913187621257642197265625
138192192512210944014297088670325762303009285989800961200925440187144214422667467682116631088149638039277531337027794364361647156637729210986328125
141638440960047964163492249617695641666169958418888629764198791168734566060810147819968110335024809342654048603976791628818273969571252411976960521913187654931640625
Technical details
Node id
n-6433a9002006e547 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-99f5d32b38d6538a

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:   1,    4,     4;
  1:   2,   11,    20,     12;
  2:   4,   28,    73,     84,      36;
  3:   8,   68,   230,    387,     324,     108;
  4:  16,  160,   664,   1464,    1809,    1188,     324;
  5:  32,  368,  1808,   4920,    8010,    7803,    4212,     972;
  6:  64,  832,  4720,  15264,   30780,   39636,   31833,   14580,    2916;
  7: 128, 1856, 11936,  44688,  107352,  171612,  182574,  124659,   49572,    8748;
  8: 256, 4096, 29440, 125184,  348768,  665280,  879984,  797040,  473121,  166212,  26244;
  9: 512, 8960, 71168, 338688, 1073088, 2376864, 3755808, 4234032, 3337362, 1751787, 551124, 78732;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2+5*x+3*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 4*x + 4*x^2.
%t Mathematica
P[0, x_] := 1 + 4*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + 5*x + 3*x^2)*P[n - 1, x])/(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 + 4*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + 5*x + 3*x^2)*v[i - 1])/(1 + x)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.