Real-rooted atlas

A126216 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2\,x + 2\,x^{2} + 6\,n + 5\,n x + n x^{2} + 5\,n^{2} + 2\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(2\,x + 2\,x^{2} + 12\,n + 18\,n x + 13\,n x^{2} + 3\,n x^{3} + 29\,n^{2} + 25\,n^{2} x + 6\,n^{2} x^{2} + 7\,n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x) + \left(-12 - 30\,x - 28\,x^{2} - 12\,x^{3} - 2\,x^{4} - 18\,n - 35\,n x - 21\,n x^{2} - 4\,n x^{3} - 4\,n^{2} - 6\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
2189128
38766151169
44394501582211070
52278295514203103016994
6120521896311640351554039146750
76466911981289838349307490868491317657
8350733748548664552317576882841439213166012185462
9191815246372054763412270832893038419782025620190872115182902
101056067828537245333247352200983090350402385348405090429902670851106303712
115846174717468371822869280517227039582440270261103925529678773190688053630991749034730
1232510409910646117821545065478130839027071625064526531564467646172118415881394415105930481811415305730956
1318148254186464582943103043322729695594766598171259425415954857601502481607188442377775423996831578726266130137749854276
141016403415239132819495679775279827038998175348357032724232098296437982514540198520393535579544045260215396996622886580081471269604206866
Technical details
Node id
n-f772ba7716013345 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7b6c44b627d65933

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       4,       1;
  2:      18,       9,       1;
  3:      87,      66,      15,       1;
  4:     439,     450,     158,      22,      1;
  5:    2278,    2955,    1420,     310,     30,      1;
  6:   12052,   18963,   11640,    3515,    540,     39,     1;
  7:   64669,  119812,   89838,   34930,   7490,    868,    49,    1;
  8:  350733,  748548,  664552,  317576,  88284,  14392,  1316,   60,  1;
  9: 1918152, 4637205, 4763412, 2708328, 930384, 197820, 25620, 1908, 72, 1;
%F Formula
The row polynomials satisfy (n^3+5*n^2+6*n+(2*n^2+5*n+2)*x+(n+2)*x^2)*P(n,x) = (7*n^3+29*n^2+12*n+(3*n^3+25*n^2+18*n+2)*x+(6*n^2+13*n+2)*x^2+3*n*x^3)*P(n-1,x) - (4*n^2+18*n+12+(6*n^2+35*n+30)*x+(2*n^2+21*n+28)*x^2+(4*n+12)*x^3+2*x^4)*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[((7*n^3 + 29*n^2 + 12*n + (3*n^3 + 25*n^2 + 18*n + 2)*x + (6*n^2 + 13*n + 2)*x^2 + 3*n*x^3)*P[n - 1, x] - (4*n^2 + 18*n + 12 + (6*n^2 + 35*n + 30)*x + (2*n^2 + 21*n + 28)*x^2 + (4*n + 12)*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n^3 + 5*n^2 + 6*n + (2*n^2 + 5*n + 2)*x + (n + 2)*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] = ((7*n^3 + 29*n^2 + 12*n + (3*n^3 + 25*n^2 + 18*n + 2)*x + (6*n^2 + 13*n + 2)*x^2 + 3*n*x^3)*v[i - 1] - (4*n^2 + 18*n + 12 + (6*n^2 + 35*n + 30)*x + (2*n^2 + 21*n + 28)*x^2 + (4*n + 12)*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n^3 + 5*n^2 + 6*n + (2*n^2 + 5*n + 2)*x + (n + 2)*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