Real-rooted atlas

A091044 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n^{2} + 3\,n^{2} x + 6\,n^{3} x\right)\,P_{n}(x) = \left(-12\,x - 9\,x^{2} + 8\,n + 12\,n x - 18\,n x^{2} + 16\,n^{2} + 84\,n^{2} x + 81\,n^{2} x^{2} + 24\,n^{3} x + 54\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 18\,x^{2} - 18\,x^{3} - 8\,n x - 48\,n x^{2} - 72\,n x^{3} - 12\,n^{2} x^{2} - 36\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141216
2169681193
36457610804322152
4256307290728640202523065
51024153606220896768567008748240808
6409673728380160829440816480326592357212466217
7163843440642156544608256085536005878656171460813996824886384
865536157286411612160402554887413120073903104377213768398080531441248191249
92621447077888601620482477260805660928007472424965604318722217093123897234019683002451645280
10104857631457280302579712144388915239481344006521389056653837184038429614081215937008173210400714492924026123761
114194304138412032148635648080687923202571927552051167821824646704414725123948544024318740160630485856074307261625509168233886959896
1216777216603979776716282265643599790080158854348800370357567488567110025216570155728896367483184640144111052800312090498723095112384898135292263849253353
1367108864261724569633973862400229210324992940120473600251625288499245368802017285555363512320457015451443224498878976008084630062081485653944321257389406031248730821804442808632
1426843545611274289152158997676032117776056320053721169920001624528178380833759726206976488871989084164921704861696033852996403200153607971179524329620066304684019836864499979692801076168025209108181033625
Technical details
Node id
n-79ec7e88b86c6479 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d93b668f42c2ba43

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,      12;
  2:     16,      96,       81;
  3:     64,     576,     1080,       432;
  4:    256,    3072,     9072,      8640,      2025;
  5:   1024,   15360,    62208,     96768,     56700,      8748;
  6:   4096,   73728,   380160,    829440,    816480,    326592,     35721;
  7:  16384,  344064,  2156544,   6082560,   8553600,   5878656,   1714608,    139968;
  8:  65536, 1572864, 11612160,  40255488,  74131200,  73903104,  37721376,   8398080,   531441;
  9: 262144, 7077888, 60162048, 247726080, 566092800, 747242496, 560431872, 221709312, 38972340, 1968300;
%F Formula
The row polynomials satisfy (4*n^2+2*n+(6*n^3+3*n^2)*x)*P(n,x) = (16*n^2+8*n+(24*n^3+84*n^2+12*n-12)*x+(54*n^3+81*n^2-18*n-9)*x^2)*P(n-1,x) - ((8*n+4)*x+(12*n^2+48*n+18)*x^2+(36*n^2+72*n+18)*x^3)*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[((16*n^2 + 8*n + (24*n^3 + 84*n^2 + 12*n - 12)*x + (54*n^3 + 81*n^2 - 18*n - 9)*x^2)*P[n - 1, x] - ((8*n + 4)*x + (12*n^2 + 48*n + 18)*x^2 + (36*n^2 + 72*n + 18)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (6*n^3 + 3*n^2)*x)]];
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] = ((16*n^2 + 8*n + (24*n^3 + 84*n^2 + 12*n - 12)*x + (54*n^3 + 81*n^2 - 18*n - 9)*x^2)*v[i - 1] - ((8*n + 4)*x + (12*n^2 + 48*n + 18)*x^2 + (36*n^2 + 72*n + 18)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (6*n^3 + 3*n^2)*x)); 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