Real-rooted atlas

A063007 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 3\,x - 4\,x^{2} - 2\,n - 3\,n x + 10\,n x^{2} - 8\,n^{2} x^{2} + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(6 + 17\,x + 12\,x^{2} - 6\,n - 13\,n x + 6\,n x^{2} + 16\,n x^{3} - 4\,n^{2} x - 24\,n^{2} x^{2} - 24\,n^{2} x^{3} + 6\,n^{3} x^{2} + 8\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-2 - 12\,x - 22\,x^{2} - 12\,x^{3} - 8\,n x^{2} - 24\,n x^{3} - 16\,n x^{4} + 4\,n^{2} x^{2} + 12\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
23811
3137272157
46352810806402311
532136001116013440560034121
61683236409828017920015120048384502387
789891517047912801935360246960015966724139527367557
8486399584966014736184665603160080030868992161441283514368107616719
9265729598838443923600162328320348717600456648192360552192158146560296524801566223057
1014625633709764031142232013459353603481077600572194022460540480004006379520151227648024893440022720574107
1180974532282979602158489080106814822403231468240063951503616849181132807527776256042847833600141892608002081091584328656614573
12450467191397469744146941938008192952768028386998640065718593740810505952737281168597647360890730846720444596838400131108769792173366968324742088234583
1325159596985168942569858390141661151073984023877886032006331800803328118412443269121585564758835215161579896320101311322112004501401096192119623208140814399089868868269680637081
141409933619517166635446534612630004463436257280193935806064005796411241420812416894422732819449503563776022386077505792018762459555840011157356309299244659331039232107993174016001192824012800980902103166083
15792384825331306255596042876336484803197765136384015302876228640050901341012121612290876050252802205283562127360296905493364480029949414663168002233481596329984119623208140800043557246853120096618745036800985905561600014068759957860373
Technical details
Node id
n-774b8b68c437bf61 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-90dff7c8d5583170

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,        8;
   3:      13,       72,        72;
   4:      63,      528,      1080,        640;
   5:     321,     3600,     11160,      13440,       5600;
   6:    1683,    23640,     98280,     179200,     151200,      48384;
   7:    8989,   151704,    791280,    1935360,    2469600,    1596672,     413952;
   8:   48639,   958496,   6014736,   18466560,   31600800,   30868992,   16144128,    3514368;
   9:  265729,  5988384,  43923600,  162328320,  348717600,  456648192,  360552192,  158146560,   29652480;
  10: 1462563, 37097640, 311422320, 1345935360, 3481077600, 5721940224, 6054048000, 4006379520, 1512276480, 248934400;
%F Formula
The row polynomials satisfy -(2*n-2+(3*n-3)*x-(2*n^3-8*n^2+10*n-4)*x^2)*P(n,x) = -(6*n-6+(4*n^2+13*n-17)*x-(6*n^3-24*n^2+6*n+12)*x^2-(8*n^3-24*n^2+16*n)*x^3)*P(n-1,x) - (2+12*x-(4*n^2-8*n-22)*x^2-(12*n^2-24*n-12)*x^3-(8*n^2-16*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 6 + (4*n^2 + 13*n - 17)*x - (6*n^3 - 24*n^2 + 6*n + 12)*x^2 - (8*n^3 - 24*n^2 + 16*n)*x^3)*P[n - 1, x] - (2 + 12*x - (4*n^2 - 8*n - 22)*x^2 - (12*n^2 - 24*n - 12)*x^3 - (8*n^2 - 16*n)*x^4)*D[P[n - 1, x], x])/(-2*n + 2 - (3*n - 3)*x + (2*n^3 - 8*n^2 + 10*n - 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(6*n - 6 + (4*n^2 + 13*n - 17)*x - (6*n^3 - 24*n^2 + 6*n + 12)*x^2 - (8*n^3 - 24*n^2 + 16*n)*x^3)*v[i - 1] - (2 + 12*x - (4*n^2 - 8*n - 22)*x^2 - (12*n^2 - 24*n - 12)*x^3 - (8*n^2 - 16*n)*x^4)*deriv(v[i - 1], x))/(-2*n + 2 - (3*n - 3)*x + (2*n^3 - 8*n^2 + 10*n - 4)*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. [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