Real-rooted atlas

A152842 → reverse → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(13 + 56\,x + 89\,x^{2} + 62\,x^{3} + 16\,x^{4} - 10\,n - 46\,n x - 78\,n x^{2} - 58\,n x^{3} - 16\,n x^{4}\right)\,P_{n-1}(x) + \left(6 + 38\,x + 94\,x^{2} + 114\,x^{3} + 68\,x^{4} + 16\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
2310821
39426432147
496014816064441
52721668410728322563087
62727011162440297619205129261
78191844281178418688176649216204864827
8811080626420640422565504044544204804096194481
92433564231128697620932833414435379223961694208163841361367
102434050302401332003832807528001022080947200573440204800327684084101
117291312210692052056016826403791520607744069299205509120290816091750413107228588707
127291458013316473440027237607156800136604801908480019368960139264006733824196608026214485766121
132187466564578122735856111088803236544069608640111896320134446080119255040759070723283353686507521048576600362847
1421875103055112436514801658059254583200134339520251113600358238720388147200314417152184647680743178241835008020971521801088541
Technical details
Node id
n-3faf35b58dcbb6d6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-765e1f4dee9a440f

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:   3,    4;
  2:   3,   10,     8;
  3:   9,   42,    64,    32;
  4:   9,   60,   148,   160,     64;
  5:  27,  216,   684,  1072,    832,    256;
  6:  27,  270,  1116,  2440,   2976,   1920,    512;
  7:  81,  918,  4428, 11784,  18688,  17664,   9216,   2048;
  8:  81, 1080,  6264, 20640,  42256,  55040,  44544,  20480,  4096;
  9: 243, 3564, 23112, 86976, 209328, 334144, 353792, 239616, 94208, 16384;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(10*n-13+(46*n-56)*x+(78*n-89)*x^2+(58*n-62)*x^3+(16*n-16)*x^4)*P(n-1,x) + (6+38*x+94*x^2+114*x^3+68*x^4+16*x^5)*d/dx 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[Together[(-(10*n - 13 + (46*n - 56)*x + (78*n - 89)*x^2 + (58*n - 62)*x^3 + (16*n - 16)*x^4)*P[n - 1, x] + (6 + 38*x + 94*x^2 + 114*x^3 + 68*x^4 + 16*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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] = (-(10*n - 13 + (46*n - 56)*x + (78*n - 89)*x^2 + (58*n - 62)*x^3 + (16*n - 16)*x^4)*v[i - 1] + (6 + 38*x + 94*x^2 + 114*x^3 + 68*x^4 + 16*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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.