Real-rooted atlas

A128100 → D → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-20 + 4\,x - 4\,x^{2} + 10\,n - 12\,n x + 8\,n x^{2} + 5\,n^{2} x - 5\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(-70 - 46\,x + 10\,x^{2} + 30\,n - 36\,n x - 41\,n x^{2} + 6\,n x^{3} + 20\,n^{2} x + 4\,n^{2} x^{2} - 8\,n^{2} x^{3} + 3\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 35\,x - 2\,x^{2} + 8\,x^{3} - 25\,n x - 40\,n x^{2} - 11\,n x^{3} + 4\,n x^{4} - 5\,n^{2} x^{2} - 9\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
211
322
4549
5101222
62036965
7388836162
87120412616417
9130444360801014
10235932945320252457
114201896230410401505810
127443768534630406753613609
1313087344119168176245025231446
142285140882574020720787512604972017
Technical details
Node id
n-66e4cb37af1e1354 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-fcd0b3f751df5a6d

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -2)
  • 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:   0;
  1:   0;
  2:   1;
  3:   2;
  4:   5,   4;
  5:  10,  12;
  6:  20,  36,   9;
  7:  38,  88,  36;
  8:  71, 204, 126, 16;
  9: 130, 444, 360, 80;
%F Formula
The row polynomials satisfy (10*n-20+(5*n^2-12*n+4)*x+(n^3-5*n^2+8*n-4)*x^2)*P(n,x) = (30*n-70+(20*n^2-36*n-46)*x+(3*n^3+4*n^2-41*n+10)*x^2+(2*n^3-8*n^2+6*n)*x^3)*P(n-1,x) - (25+(25*n+35)*x+(5*n^2+40*n+2)*x^2+(9*n^2+11*n-8)*x^3+(4*n^2-4*n)*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((30*n - 70 + (20*n^2 - 36*n - 46)*x + (3*n^3 + 4*n^2 - 41*n + 10)*x^2 + (2*n^3 - 8*n^2 + 6*n)*x^3)*P[n - 1, x] - (25 + (25*n + 35)*x + (5*n^2 + 40*n + 2)*x^2 + (9*n^2 + 11*n - 8)*x^3 + (4*n^2 - 4*n)*x^4)*D[P[n - 1, x], x])/(10*n - 20 + (5*n^2 - 12*n + 4)*x + (n^3 - 5*n^2 + 8*n - 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((30*n - 70 + (20*n^2 - 36*n - 46)*x + (3*n^3 + 4*n^2 - 41*n + 10)*x^2 + (2*n^3 - 8*n^2 + 6*n)*x^3)*v[i - 1] - (25 + (25*n + 35)*x + (5*n^2 + 40*n + 2)*x^2 + (9*n^2 + 11*n - 8)*x^3 + (4*n^2 - 4*n)*x^4)*deriv(v[i - 1], x))/(10*n - 20 + (5*n^2 - 12*n + 4)*x + (n^3 - 5*n^2 + 8*n - 4)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.