Real-rooted atlas

A143495 → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-6 - 34\,x - 76\,x^{2} - 83\,x^{3} - 44\,x^{4} - 9\,x^{5} + 2\,n x + 9\,n x^{2} + 15\,n x^{3} + 11\,n x^{4} + 3\,n x^{5}\right)\,P_{n-2}(x) + \left(-2\,x - 11\,x^{2} - 24\,x^{3} - 26\,x^{4} - 14\,x^{5} - 3\,x^{6}\right)\,P_{n-2}'(x) + \left(5 + 19\,x + 25\,x^{2} + 13\,x^{3} + 2\,x^{4} - n x - 3\,n x^{2} - 3\,n x^{3} - n x^{4}\right)\,P_{n-1}(x) + \left(x + 4\,x^{2} + 6\,x^{3} + 4\,x^{4} + x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2358
39322667
427155265141588
581674182720287995409
62432777106581856215168473652144
7729111085684813845217631311369429371526515
8218743703287607920719162707116272158652611904975564260
96561170438140852957045841313740018134638149000636733904129102061487137
101968366206967607443376402897440325174251795196813242137030402538055659131275709679128
11590492569544320599741938994086828640401520450766221500149321131771201275144252442342806673108478544879299
1217714799829311510112211091577313460289675812419670738224206731082749384143222673302452120621424243745523978516369838107187169340
1353144138868842709158607606682389230200495053968376571882107730865383191109604403373135847652447065360601163246264543267705322041164167971398875799297
1415943231517552813328825758334560891101944932811107303070307541878778326696341619798507244452602720094124013593715266833433307811457304842902937114702193405116498518967816206400
1547829695942665401564735636418372591404412368317111315377028918102161060305459073447243617988053706434774437611877350221125053053762472629484742394364115378659715052718961471664291871399682266850448333427
Technical details
Node id
n-c9fef5954ad29cf9 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0cf40a9b7a537b20

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 -1)
  • 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     3,      5;
   3:     9,     32,      26;
   4:    27,    155,     265,      141;
   5:    81,    674,    1827,     2028,      799;
   6:   243,   2777,   10658,    18562,    15168,      4736;
   7:   729,  11108,   56848,   138452,   176313,    113694,     29371;
   8:  2187,  43703,  287607,   920719,  1627071,   1627215,    865261,    190497;
   9:  6561, 170438, 1408529,  5704584, 13137400,  18134638,  14900063,   6733904,  1291020;
  10: 19683, 662069, 6760744, 33764028, 97440325, 174251795, 196813242, 137030402, 53805565, 9131275;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(6-(2*n-34)*x-(9*n-76)*x^2-(15*n-83)*x^3-(11*n-44)*x^4-(3*n-9)*x^5)*P(n-2,x) - (2*x+11*x^2+24*x^3+26*x^4+14*x^5+3*x^6)*d/dx P(n-2,x) + (5-(n-19)*x-(3*n-25)*x^2-(3*n-13)*x^3-(n-2)*x^4)*P(n-1,x) + (x+4*x^2+6*x^3+4*x^4+x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 5*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(6 - (2*n - 34)*x - (9*n - 76)*x^2 - (15*n - 83)*x^3 - (11*n - 44)*x^4 - (3*n - 9)*x^5)*P[n - 2, x] - (2*x + 11*x^2 + 24*x^3 + 26*x^4 + 14*x^5 + 3*x^6)*D[P[n - 2, x], x] + (5 - (n - 19)*x - (3*n - 25)*x^2 - (3*n - 13)*x^3 - (n - 2)*x^4)*P[n - 1, x] + (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 5*x; for(n = 3, N, my(i = n); v[i] = (-(6 - (2*n - 34)*x - (9*n - 76)*x^2 - (15*n - 83)*x^3 - (11*n - 44)*x^4 - (3*n - 9)*x^5)*v[i - 2] - (2*x + 11*x^2 + 24*x^3 + 26*x^4 + 14*x^5 + 3*x^6)*deriv(v[i - 2], x) + (5 - (n - 19)*x - (3*n - 25)*x^2 - (3*n - 13)*x^3 - (n - 2)*x^4)*v[i - 1] + (x + 4*x^2 + 6*x^3 + 4*x^4 + 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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.