Real-rooted atlas

A008292 → γ (scale 1/2) (shift +1) → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-x + n x^{3}\right)\,P_{n}(x) = \left(-2 - 8\,x - 6\,x^{2} + 8\,x^{3} + 8\,x^{4} + 2\,n + 4\,n x + 2\,n x^{3} + 8\,n x^{4} + 2\,n x^{5} + n^{2} x - 4\,n^{2} x^{3} - 2\,n^{2} x^{4} + n^{2} x^{5} - n^{3} x^{3} - 2\,n^{3} x^{4} - n^{3} x^{5}\right)\,P_{n-2}(x) + \left(-1 - 5\,x - 7\,x^{2} + x^{3} + 8\,x^{4} + 4\,x^{5} - n x - 3\,n x^{2} + n x^{3} + 10\,n x^{4} + 9\,n x^{5} + 2\,n x^{6} + n^{2} x^{3} + 4\,n^{2} x^{4} + 5\,n^{2} x^{5} + 2\,n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(-2 - 2\,x + 4\,x^{2} + 2\,x^{3} - 2\,n x + 3\,n x^{3} - 2\,n x^{4} + 2\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 2\,x^{3} - n x^{3} - 3\,n x^{4} - 2\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2224
35813
416381266
561188102351
627210388461362292
7138562886768198416425

All 15 rows

Technical details
Node id
n-d5117d09cd06dd68 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c2e210e058214845

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:     1;
  2:     2,      2;
  3:     5,      8;
  4:    16,     38,     12;
  5:    61,    188,    102;
  6:   272,   1038,    846,    136;
  7:  1385,   6288,   6768,   1984;
  8:  7936,  41806,  56232,  25088,  2480;
  9: 50521, 302836, 488502, 291968, 55280;
%F Formula
The row polynomials satisfy -(x-n*x^3)*P(n,x) = (2*n-2+(n^2+4*n-8)*x-6*x^2-(n^3+4*n^2-2*n-8)*x^3-(2*n^3+2*n^2-8*n-8)*x^4-(n^3-n^2-2*n)*x^5)*P(n-2,x) - (1+(n+5)*x+(3*n+7)*x^2-(n^2+n+1)*x^3-(4*n^2+10*n+8)*x^4-(5*n^2+9*n+4)*x^5-(2*n^2+2*n)*x^6)*d/dx P(n-2,x) - (2+(2*n+2)*x-4*x^2-(2*n^2+3*n+2)*x^3-(2*n^2-2*n)*x^4)*P(n-1,x) + (x+3*x^2-(n-2)*x^3-3*n*x^4-2*n*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n^2 + 4*n - 8)*x - 6*x^2 - (n^3 + 4*n^2 - 2*n - 8)*x^3 - (2*n^3 + 2*n^2 - 8*n - 8)*x^4 - (n^3 - n^2 - 2*n)*x^5)*P[n - 2, x] - (1 + (n + 5)*x + (3*n + 7)*x^2 - (n^2 + n + 1)*x^3 - (4*n^2 + 10*n + 8)*x^4 - (5*n^2 + 9*n + 4)*x^5 - (2*n^2 + 2*n)*x^6)*D[P[n - 2, x], x] - (2 + (2*n + 2)*x - 4*x^2 - (2*n^2 + 3*n + 2)*x^3 - (2*n^2 - 2*n)*x^4)*P[n - 1, x] + (x + 3*x^2 - (n - 2)*x^3 - 3*n*x^4 - 2*n*x^5)*D[P[n - 1, x], x])/(-x + n*x^3)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n^2 + 4*n - 8)*x - 6*x^2 - (n^3 + 4*n^2 - 2*n - 8)*x^3 - (2*n^3 + 2*n^2 - 8*n - 8)*x^4 - (n^3 - n^2 - 2*n)*x^5)*v[i - 2] - (1 + (n + 5)*x + (3*n + 7)*x^2 - (n^2 + n + 1)*x^3 - (4*n^2 + 10*n + 8)*x^4 - (5*n^2 + 9*n + 4)*x^5 - (2*n^2 + 2*n)*x^6)*deriv(v[i - 2], x) - (2 + (2*n + 2)*x - 4*x^2 - (2*n^2 + 3*n + 2)*x^3 - (2*n^2 - 2*n)*x^4)*v[i - 1] + (x + 3*x^2 - (n - 2)*x^3 - 3*n*x^4 - 2*n*x^5)*deriv(v[i - 1], x))/(-x + n*x^3)); 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.