Real-rooted atlas

A367270 → basis A013611 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-5\,n + 15\,n x - 4\,n x^{2} + 15\,n^{2} - 23\,n^{2} x + 4\,n^{2} x^{2} - 14\,n^{3} + 8\,n^{3} x + 4\,n^{4}\right)\,P_{n}(x) = \left(-6 - 46\,x - 80\,x^{2} + 32\,x^{3} - 2\,n + 66\,n x + 260\,n x^{2} - 80\,n x^{3} + 40\,n^{2} + 68\,n^{2} x - 276\,n^{2} x^{2} + 48\,n^{2} x^{3} - 48\,n^{3} - 136\,n^{3} x + 96\,n^{3} x^{2} + 16\,n^{4} + 48\,n^{4} x\right)\,P_{n-1}(x) + \left(-2\,x - 15\,x^{2} - 24\,x^{3} + 16\,x^{4} - 2\,n x + 14\,n x^{2} + 80\,n x^{3} - 32\,n x^{4} + 12\,n^{2} x + 32\,n^{2} x^{2} - 64\,n^{2} x^{3} - 8\,n^{3} x - 32\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
211819
35814448250
4275120012482562979
512467840148808960128034206
6550245360127680143360576006144385646
72389224393691728015590401182720344064286724299604
81025311249248589881613601280158592008773632194969613107247565475
94361506177600351711361030149121649894401419264006021120010616832589824523133494
101842698297554401985068807072665601452971520173434060811543347203892838405603328026214405726956986
11774295614041456010746964804518159360113777664001756481126416359383040871563264024005836802883584001153433662459083116
1232391086651819168563136288027314327040816629299201555740426241887356190721416993177606199345152014244904960145332633650331648679043824014
1313499978829853882242874212140815809201408054806019840012463846686721876925546496184681444147211454028185604198498304008187071692871974256642181038087362678273900
145608895001352078000014355823372888332878233634871290854409237208780800166865707008002048702231347216741774393344874337271808027290238976004580179968003511471308893952409679647142809084
Technical details
Node id
n-414d59e057d30255 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3a1ad01786490d10

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      1;
  2:     11,       8;
  3:     58,     144,       48;
  4:    275,    1200,     1248,       256;
  5:   1246,    7840,    14880,      8960,      1280;
  6:   5502,   45360,   127680,    143360,     57600,      6144;
  7:  23892,  243936,   917280,   1559040,   1182720,    344064,    28672;
  8: 102531, 1249248,  5898816,  13601280,  15859200,   8773632,  1949696,   131072;
  9: 436150, 6177600, 35171136, 103014912, 164989440, 141926400, 60211200, 10616832, 589824;
%F Formula
The row polynomials satisfy (4*n^4-14*n^3+15*n^2-5*n+(8*n^3-23*n^2+15*n)*x+(4*n^2-4*n)*x^2)*P(n,x) = (16*n^4-48*n^3+40*n^2-2*n-6+(48*n^4-136*n^3+68*n^2+66*n-46)*x+(96*n^3-276*n^2+260*n-80)*x^2+(48*n^2-80*n+32)*x^3)*P(n-1,x) - ((8*n^3-12*n^2+2*n+2)*x+(32*n^3-32*n^2-14*n+15)*x^2+(64*n^2-80*n+24)*x^3+(32*n-16)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^4 - 48*n^3 + 40*n^2 - 2*n - 6 + (48*n^4 - 136*n^3 + 68*n^2 + 66*n - 46)*x + (96*n^3 - 276*n^2 + 260*n - 80)*x^2 + (48*n^2 - 80*n + 32)*x^3)*P[n - 1, x] - ((8*n^3 - 12*n^2 + 2*n + 2)*x + (32*n^3 - 32*n^2 - 14*n + 15)*x^2 + (64*n^2 - 80*n + 24)*x^3 + (32*n - 16)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 14*n^3 + 15*n^2 - 5*n + (8*n^3 - 23*n^2 + 15*n)*x + (4*n^2 - 4*n)*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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^4 - 48*n^3 + 40*n^2 - 2*n - 6 + (48*n^4 - 136*n^3 + 68*n^2 + 66*n - 46)*x + (96*n^3 - 276*n^2 + 260*n - 80)*x^2 + (48*n^2 - 80*n + 32)*x^3)*v[i - 1] - ((8*n^3 - 12*n^2 + 2*n + 2)*x + (32*n^3 - 32*n^2 - 14*n + 15)*x^2 + (64*n^2 - 80*n + 24)*x^3 + (32*n - 16)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 14*n^3 + 15*n^2 - 5*n + (8*n^3 - 23*n^2 + 15*n)*x + (4*n^2 - 4*n)*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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023