Real-rooted atlas

A099093 → θ+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 6\,n x + 12\,n^{2} - 27\,n^{2} x + 13\,n^{3} - 36\,n^{3} x + 9\,n^{3} x^{2} + 6\,n^{4} - 9\,n^{4} x + n^{5}\right)\,P_{n}(x) = \left(8 + 108\,x - 324\,x^{2} + 54\,x^{3} + 16\,n + 264\,n x - 702\,n x^{2} + 108\,n x^{3} + 2\,n^{2} + 300\,n^{2} x - 459\,n^{2} x^{2} + 54\,n^{2} x^{3} - 14\,n^{3} + 162\,n^{3} x - 99\,n^{3} x^{2} - 10\,n^{4} + 30\,n^{4} x - 2\,n^{5}\right)\,P_{n-1}(x) + \left(16\,x - 108\,x^{2} - 54\,x^{3} + 27\,x^{4} + 48\,n x - 228\,n x^{2} - 126\,n x^{3} + 54\,n x^{4} + 52\,n^{2} x - 141\,n^{2} x^{2} - 99\,n^{2} x^{3} + 27\,n^{2} x^{4} + 24\,n^{3} x - 18\,n^{3} x^{2} - 27\,n^{3} x^{3} + 4\,n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
102424
2024243267
30048617282214
40024351841012515552
50005184405005248898172
6000172860750262440250047574965
7000040500524880150028211197443185406

All 15 rows

Technical details
Node id
n-e9dc8707d149685a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-514675ebedd78e4b

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: 0, 24;
  2: 0, 24, 243;
  3: 0,  0, 486, 1728;
  4: 0,  0, 243, 5184, 10125;
  5: 0,  0,   0, 5184, 40500,  52488;
  6: 0,  0,   0, 1728, 60750, 262440,  250047;
  7: 0,  0,   0,    0, 40500, 524880, 1500282,  1119744;
  8: 0,  0,   0,    0, 10125, 524880, 3750705,  7838208,  4782969;
  9: 0,  0,   0,    0,     0, 262440, 5000940, 23514624, 38263752, 19683000;
%F Formula
The row polynomials satisfy (n^5+6*n^4+13*n^3+12*n^2+4*n-(9*n^4+36*n^3+27*n^2-6*n)*x+9*n^3*x^2)*P(n,x) = -(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8-(30*n^4+162*n^3+300*n^2+264*n+108)*x+(99*n^3+459*n^2+702*n+324)*x^2-(54*n^2+108*n+54)*x^3)*P(n-1,x) + ((4*n^4+24*n^3+52*n^2+48*n+16)*x+(3*n^4-18*n^3-141*n^2-228*n-108)*x^2-(27*n^3+99*n^2+126*n+54)*x^3+(27*n^2+54*n+27)*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[(-(2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8 - (30*n^4 + 162*n^3 + 300*n^2 + 264*n + 108)*x + (99*n^3 + 459*n^2 + 702*n + 324)*x^2 - (54*n^2 + 108*n + 54)*x^3)*P[n - 1, x] + ((4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x + (3*n^4 - 18*n^3 - 141*n^2 - 228*n - 108)*x^2 - (27*n^3 + 99*n^2 + 126*n + 54)*x^3 + (27*n^2 + 54*n + 27)*x^4)*D[P[n - 1, x], x])/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n - (9*n^4 + 36*n^3 + 27*n^2 - 6*n)*x + 9*n^3*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] = (-(2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8 - (30*n^4 + 162*n^3 + 300*n^2 + 264*n + 108)*x + (99*n^3 + 459*n^2 + 702*n + 324)*x^2 - (54*n^2 + 108*n + 54)*x^3)*v[i - 1] + ((4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x + (3*n^4 - 18*n^3 - 141*n^2 - 228*n - 108)*x^2 - (27*n^3 + 99*n^2 + 126*n + 54)*x^3 + (27*n^2 + 54*n + 27)*x^4)*deriv(v[i - 1], x))/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n - (9*n^4 + 36*n^3 + 27*n^2 - 6*n)*x + 9*n^3*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. [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
  3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.