Real-rooted atlas

A081579 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(80 - 40\,x - 28\,x^{2} - 14\,x^{3} + 2\,x^{4} + 240\,n x - 20\,n x^{2} - 34\,n x^{3} - 4\,n x^{4} + 220\,n^{2} x^{2} + 50\,n^{2} x^{3} - 5\,n^{2} x^{4} - n^{2} x^{5} + 60\,n^{3} x^{3} + 10\,n^{3} x^{4} - n^{3} x^{5} + 5\,n^{4} x^{4}\right)\,P_{n}(x) = \left(320 + 1120\,x + 1488\,x^{2} + 936\,x^{3} + 280\,x^{4} + 24\,x^{5} - 8\,x^{6} + 960\,n x + 2640\,n x^{2} + 2664\,n x^{3} + 1128\,n x^{4} + 124\,n x^{5} - 20\,n x^{6} + 880\,n^{2} x^{2} + 1800\,n^{2} x^{3} + 1140\,n^{2} x^{4} + 204\,n^{2} x^{5} - 16\,n^{2} x^{6} + 240\,n^{3} x^{3} + 360\,n^{3} x^{4} + 116\,n^{3} x^{5} - 4\,n^{3} x^{6} + 20\,n^{4} x^{4} + 20\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(160 + 240\,x + 64\,x^{2} - 120\,x^{3} - 54\,x^{4} - x^{5} - x^{6} + 480\,n x + 640\,n x^{2} + 312\,n x^{3} + 28\,n x^{4} - 4\,n x^{5} - 3\,n x^{6} + 440\,n^{2} x^{2} + 500\,n^{2} x^{3} + 180\,n^{2} x^{4} + 10\,n^{2} x^{5} - 3\,n^{2} x^{6} + 120\,n^{3} x^{3} + 100\,n^{3} x^{4} + 18\,n^{3} x^{5} - n^{3} x^{6} + 10\,n^{4} x^{4} + 5\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
2832949
32414412616310
480640900320251965
52562560504032006503611742
683299842520023040840011524968657
72688376321160641395207540018288186264391518

All 15 rows

Technical details
Node id
n-a05bfb4e8ba2d117 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-bd6cfc3f76828922

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:     2,      4;
  2:     8,     32,       9;
  3:    24,    144,     126,      16;
  4:    80,    640,     900,     320,      25;
  5:   256,   2560,    5040,    3200,     650,      36;
  6:   832,   9984,   25200,   23040,    8400,    1152,     49;
  7:  2688,  37632,  116064,  139520,   75400,   18288,   1862,    64;
  8:  8704, 139264,  507456,  756736,  546400,  198144,  35084,  2816,   81;
  9: 28160, 506880, 2131200, 3799040, 3448800, 1690560, 446880, 61440, 4050, 100;
%F Formula
The row polynomials satisfy (80+(240*n-40)*x+(220*n^2-20*n-28)*x^2+(60*n^3+50*n^2-34*n-14)*x^3+(5*n^4+10*n^3-5*n^2-4*n+2)*x^4-(n^3+n^2)*x^5)*P(n,x) = (320+(960*n+1120)*x+(880*n^2+2640*n+1488)*x^2+(240*n^3+1800*n^2+2664*n+936)*x^3+(20*n^4+360*n^3+1140*n^2+1128*n+280)*x^4+(20*n^4+116*n^3+204*n^2+124*n+24)*x^5-(4*n^3+16*n^2+20*n+8)*x^6)*P(n-2,x) + (160+(480*n+240)*x+(440*n^2+640*n+64)*x^2+(120*n^3+500*n^2+312*n-120)*x^3+(10*n^4+100*n^3+180*n^2+28*n-54)*x^4+(5*n^4+18*n^3+10*n^2-4*n-1)*x^5-(n^3+3*n^2+3*n+1)*x^6)*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[((320 + (960*n + 1120)*x + (880*n^2 + 2640*n + 1488)*x^2 + (240*n^3 + 1800*n^2 + 2664*n + 936)*x^3 + (20*n^4 + 360*n^3 + 1140*n^2 + 1128*n + 280)*x^4 + (20*n^4 + 116*n^3 + 204*n^2 + 124*n + 24)*x^5 - (4*n^3 + 16*n^2 + 20*n + 8)*x^6)*P[n - 2, x] + (160 + (480*n + 240)*x + (440*n^2 + 640*n + 64)*x^2 + (120*n^3 + 500*n^2 + 312*n - 120)*x^3 + (10*n^4 + 100*n^3 + 180*n^2 + 28*n - 54)*x^4 + (5*n^4 + 18*n^3 + 10*n^2 - 4*n - 1)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*P[n - 1, x])/(80 + (240*n - 40)*x + (220*n^2 - 20*n - 28)*x^2 + (60*n^3 + 50*n^2 - 34*n - 14)*x^3 + (5*n^4 + 10*n^3 - 5*n^2 - 4*n + 2)*x^4 - (n^3 + n^2)*x^5)]];
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] = ((320 + (960*n + 1120)*x + (880*n^2 + 2640*n + 1488)*x^2 + (240*n^3 + 1800*n^2 + 2664*n + 936)*x^3 + (20*n^4 + 360*n^3 + 1140*n^2 + 1128*n + 280)*x^4 + (20*n^4 + 116*n^3 + 204*n^2 + 124*n + 24)*x^5 - (4*n^3 + 16*n^2 + 20*n + 8)*x^6)*v[i - 2] + (160 + (480*n + 240)*x + (440*n^2 + 640*n + 64)*x^2 + (120*n^3 + 500*n^2 + 312*n - 120)*x^3 + (10*n^4 + 100*n^3 + 180*n^2 + 28*n - 54)*x^4 + (5*n^4 + 18*n^3 + 10*n^2 - 4*n - 1)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*v[i - 1])/(80 + (240*n - 40)*x + (220*n^2 - 20*n - 28)*x^2 + (60*n^3 + 50*n^2 - 34*n - 14)*x^3 + (5*n^4 + 10*n^3 - 5*n^2 - 4*n + 2)*x^4 - (n^3 + n^2)*x^5)); 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.