Real-rooted atlas

A119808 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(2 + 17\,x + 55\,x^{2} - 16\,x^{3} + 8\,x^{4} + n x - 14\,n x^{2} - 72\,n x^{3} + 24\,n x^{4} - 11\,n^{2} x^{2} - 35\,n^{2} x^{3} + 26\,n^{2} x^{4} + n^{3} x^{3} + 12\,n^{3} x^{4} + 2\,n^{4} x^{4}\right)\,P_{n}(x) = \left(-4 - 32\,x - 24\,x^{2} + 304\,x^{3} - 388\,x^{4} + 144\,x^{5} - 2\,n x + 74\,n x^{2} + 206\,n x^{3} - 518\,n x^{4} + 240\,n x^{5} + 22\,n^{2} x^{2} + 42\,n^{2} x^{3} - 212\,n^{2} x^{4} + 148\,n^{2} x^{5} - 2\,n^{3} x^{3} - 38\,n^{3} x^{4} + 40\,n^{3} x^{5} - 4\,n^{4} x^{4} + 4\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(6 + 58\,x + 204\,x^{2} + 40\,x^{3} + 16\,x^{4} + 3\,n x - 75\,n x^{2} - 336\,n x^{3} + 60\,n x^{4} - 33\,n^{2} x^{2} - 106\,n^{2} x^{3} + 106\,n^{2} x^{4} + 3\,n^{3} x^{3} + 48\,n^{3} x^{4} + 6\,n^{4} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4Pn(1)
011
1246
242024
38681894
416196144356
532516702641314
664128427007044752
712830769018448020016902

All 15 rows

Technical details
Node id
n-4489b3eceb351b2e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c7ad194f9e6f30de

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   2,     4;
  2:   4,    20;
  3:   8,    68,    18;
  4:  16,   196,   144;
  5:  32,   516,   702,    64;
  6:  64,  1284,  2700,   704;
  7: 128,  3076,  9018,  4480,   200;
  8: 256,  7172, 27432, 21632,  2800;
  9: 512, 16388, 78102, 88000, 22000, 576;
%F Formula
The row polynomials satisfy (2+(n+17)*x-(11*n^2+14*n-55)*x^2+(n^3-35*n^2-72*n-16)*x^3+(2*n^4+12*n^3+26*n^2+24*n+8)*x^4)*P(n,x) = -(4+(2*n+32)*x-(22*n^2+74*n-24)*x^2+(2*n^3-42*n^2-206*n-304)*x^3+(4*n^4+38*n^3+212*n^2+518*n+388)*x^4-(4*n^4+40*n^3+148*n^2+240*n+144)*x^5)*P(n-2,x) + (6+(3*n+58)*x-(33*n^2+75*n-204)*x^2+(3*n^3-106*n^2-336*n+40)*x^3+(6*n^4+48*n^3+106*n^2+60*n+16)*x^4)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4 + (2*n + 32)*x - (22*n^2 + 74*n - 24)*x^2 + (2*n^3 - 42*n^2 - 206*n - 304)*x^3 + (4*n^4 + 38*n^3 + 212*n^2 + 518*n + 388)*x^4 - (4*n^4 + 40*n^3 + 148*n^2 + 240*n + 144)*x^5)*P[n - 2, x] + (6 + (3*n + 58)*x - (33*n^2 + 75*n - 204)*x^2 + (3*n^3 - 106*n^2 - 336*n + 40)*x^3 + (6*n^4 + 48*n^3 + 106*n^2 + 60*n + 16)*x^4)*P[n - 1, x])/(2 + (n + 17)*x - (11*n^2 + 14*n - 55)*x^2 + (n^3 - 35*n^2 - 72*n - 16)*x^3 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4 + (2*n + 32)*x - (22*n^2 + 74*n - 24)*x^2 + (2*n^3 - 42*n^2 - 206*n - 304)*x^3 + (4*n^4 + 38*n^3 + 212*n^2 + 518*n + 388)*x^4 - (4*n^4 + 40*n^3 + 148*n^2 + 240*n + 144)*x^5)*v[i - 2] + (6 + (3*n + 58)*x - (33*n^2 + 75*n - 204)*x^2 + (3*n^3 - 106*n^2 - 336*n + 40)*x^3 + (6*n^4 + 48*n^3 + 106*n^2 + 60*n + 16)*x^4)*v[i - 1])/(2 + (n + 17)*x - (11*n^2 + 14*n - 55)*x^2 + (n^3 - 35*n^2 - 72*n - 16)*x^3 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^4)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.