Real-rooted atlas

A111125 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(90 + 108\,x - 162\,x^{2} - 60\,n - 72\,n x + 351\,n x^{2} - 243\,n^{2} x^{2} + 54\,n^{3} x^{2}\right)\,P_{n}(x) = \left(60 + 162\,x + 108\,x^{2} - 90\,n - 153\,n x + 108\,n x^{2} + 162\,n x^{3} - 45\,n^{2} x - 297\,n^{2} x^{2} - 243\,n^{2} x^{3} + 81\,n^{3} x^{2} + 81\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 150\,x - 261\,x^{2} - 108\,x^{3} - 90\,n x^{2} - 324\,n x^{3} - 162\,n x^{4} + 45\,n^{2} x^{2} + 162\,n^{2} x^{3} + 81\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
24610
311422780
429186270108593
576690162014044054195
619923407695105846480145828756
75217524319956112858725277025103192698
8136423358122148300780398925293058112266174961269395
93571707344395601329804226597522876021357398437400590498251093
109349210270151483554468721138576514727258119665355948640165337219683053059726
112447661617050511602107015252317090828829268608761058436640249777276101730649539338215220
1264079178509616411410779682242245785754222630445363865364645188002705034691013674502208432621257642140056773
131677615123592522160022784801609137994751992389076300099265231741593122354283630119967885040011602478653268690873313456968535
14439204145922461632927609663362643561120450884458229415439953186193470768001750513509911345281200513720395115433046642763493202232052284166987960
151149851412883705033273853273699672133933228653736139142674274904935107804053440115457505867911866150805245092378921363928200583787938595978244671744535523981517246
Technical details
Node id
n-5f48e4f1f9f7fc7a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3ebf8f974508976d

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:    4,      6;
   3:   11,     42,      27;
   4:   29,    186,     270,     108;
   5:   76,    690,    1620,    1404,      405;
   6:  199,   2340,    7695,   10584,     6480,     1458;
   7:  521,   7524,   31995,   61128,    58725,    27702,     5103;
   8: 1364,  23358,  122148,  300780,   398925,   293058,   112266,   17496;
   9: 3571,  70734,  439560, 1329804,  2265975,  2287602,  1357398,  437400,   59049;
  10: 9349, 210270, 1514835, 5446872, 11385765, 14727258, 11966535, 5948640, 1653372, 196830;
%F Formula
The row polynomials satisfy -(60*n-90+(72*n-108)*x-(54*n^3-243*n^2+351*n-162)*x^2)*P(n,x) = -(90*n-60+(45*n^2+153*n-162)*x-(81*n^3-297*n^2+108*n+108)*x^2-(81*n^3-243*n^2+162*n)*x^3)*P(n-1,x) - (25+150*x-(45*n^2-90*n-261)*x^2-(162*n^2-324*n-108)*x^3-(81*n^2-162*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[(-(90*n - 60 + (45*n^2 + 153*n - 162)*x - (81*n^3 - 297*n^2 + 108*n + 108)*x^2 - (81*n^3 - 243*n^2 + 162*n)*x^3)*P[n - 1, x] - (25 + 150*x - (45*n^2 - 90*n - 261)*x^2 - (162*n^2 - 324*n - 108)*x^3 - (81*n^2 - 162*n)*x^4)*D[P[n - 1, x], x])/(-60*n + 90 - (72*n - 108)*x + (54*n^3 - 243*n^2 + 351*n - 162)*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] = (-(90*n - 60 + (45*n^2 + 153*n - 162)*x - (81*n^3 - 297*n^2 + 108*n + 108)*x^2 - (81*n^3 - 243*n^2 + 162*n)*x^3)*v[i - 1] - (25 + 150*x - (45*n^2 - 90*n - 261)*x^2 - (162*n^2 - 324*n - 108)*x^3 - (81*n^2 - 162*n)*x^4)*deriv(v[i - 1], x))/(-60*n + 90 - (72*n - 108)*x + (54*n^3 - 243*n^2 + 351*n - 162)*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