Real-rooted atlas

A063967 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-8\,x^{2} - 10\,x^{3} - n + n x + 22\,n x^{2} + 25\,n x^{3} + n^{2} - 3\,n^{2} x - 20\,n^{2} x^{2} - 20\,n^{2} x^{3} + 2\,n^{3} x + 6\,n^{3} x^{2} + 5\,n^{3} x^{3}\right)\,P_{n}(x) = \left(2\,x^{2} + 16\,x^{3} + 30\,x^{4} - n - 13\,n x - 43\,n x^{2} - 58\,n x^{3} - 5\,n x^{4} + 50\,n x^{5} + n^{2} + 5\,n^{2} x - 3\,n^{2} x^{2} - 51\,n^{2} x^{3} - 105\,n^{2} x^{4} - 75\,n^{2} x^{5} + 2\,n^{3} x + 14\,n^{3} x^{2} + 39\,n^{3} x^{3} + 50\,n^{3} x^{4} + 25\,n^{3} x^{5}\right)\,P_{n-1}(x) + \left(x^{2} + 9\,x^{3} + 23\,x^{4} + 15\,x^{5} - n x - 9\,n x^{2} - 23\,n x^{3} - 10\,n x^{4} + 30\,n x^{5} + 25\,n x^{6} - 2\,n^{2} x^{2} - 18\,n^{2} x^{3} - 51\,n^{2} x^{4} - 60\,n^{2} x^{5} - 25\,n^{2} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
316613
4110211244
5114486025148
61188717215048476
7122138380520348911500
8126201716135514107631684640

All 15 rows

Technical details
Node id
n-fcb6e72a316133e7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-df3d0e827dd81baa

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  6,   6;
   4: 1, 10,  21,   12;
   5: 1, 14,  48,   60,   25;
   6: 1, 18,  87,  172,  150,    48;
   7: 1, 22, 138,  380,  520,   348,    91;
   8: 1, 26, 201,  716, 1355,  1410,   763,  168;
   9: 1, 30, 276, 1212, 2955,  4230,  3542, 1608,  306;
  10: 1, 34, 363, 1900, 5700, 10476, 12019, 8408, 3285, 550;
%F Formula
The row polynomials satisfy (n^2-n+(2*n^3-3*n^2+n)*x+(6*n^3-20*n^2+22*n-8)*x^2+(5*n^3-20*n^2+25*n-10)*x^3)*P(n,x) = (n^2-n+(2*n^3+5*n^2-13*n)*x+(14*n^3-3*n^2-43*n+2)*x^2+(39*n^3-51*n^2-58*n+16)*x^3+(50*n^3-105*n^2-5*n+30)*x^4+(25*n^3-75*n^2+50*n)*x^5)*P(n-1,x) - (n*x+(2*n^2+9*n-1)*x^2+(18*n^2+23*n-9)*x^3+(51*n^2+10*n-23)*x^4+(60*n^2-30*n-15)*x^5+(25*n^2-25*n)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - n + (2*n^3 + 5*n^2 - 13*n)*x + (14*n^3 - 3*n^2 - 43*n + 2)*x^2 + (39*n^3 - 51*n^2 - 58*n + 16)*x^3 + (50*n^3 - 105*n^2 - 5*n + 30)*x^4 + (25*n^3 - 75*n^2 + 50*n)*x^5)*P[n - 1, x] - (n*x + (2*n^2 + 9*n - 1)*x^2 + (18*n^2 + 23*n - 9)*x^3 + (51*n^2 + 10*n - 23)*x^4 + (60*n^2 - 30*n - 15)*x^5 + (25*n^2 - 25*n)*x^6)*D[P[n - 1, x], x])/(n^2 - n + (2*n^3 - 3*n^2 + n)*x + (6*n^3 - 20*n^2 + 22*n - 8)*x^2 + (5*n^3 - 20*n^2 + 25*n - 10)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^2 - n + (2*n^3 + 5*n^2 - 13*n)*x + (14*n^3 - 3*n^2 - 43*n + 2)*x^2 + (39*n^3 - 51*n^2 - 58*n + 16)*x^3 + (50*n^3 - 105*n^2 - 5*n + 30)*x^4 + (25*n^3 - 75*n^2 + 50*n)*x^5)*v[i - 1] - (n*x + (2*n^2 + 9*n - 1)*x^2 + (18*n^2 + 23*n - 9)*x^3 + (51*n^2 + 10*n - 23)*x^4 + (60*n^2 - 30*n - 15)*x^5 + (25*n^2 - 25*n)*x^6)*deriv(v[i - 1], x))/(n^2 - n + (2*n^3 - 3*n^2 + n)*x + (6*n^3 - 20*n^2 + 22*n - 8)*x^2 + (5*n^3 - 20*n^2 + 25*n - 10)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.