Real-rooted atlas

A008278 → x ↦ x+1 (shift +1) → θ+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2510318
3154630495
452218228765579
52031088159389618063966
687757521092388163155414729944
7414032226756248035642355104949388246141

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-e5c179e7deb9c036 · revision 1 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5f72f3d1614d5d16

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      2,       2;
  2:      5,      10,       3;
  3:     15,      46,      30,       4;
  4:     52,     218,     228,      76,       5;
  5:    203,    1088,    1593,     896,     180,       6;
  6:    877,    5752,   10923,    8816,    3155,     414,      7;
  7:   4140,   32226,   75624,   80356,   42355,   10494,    938,      8;
  8:  21147,  190990,  535692,  710524,  500855,  187986,  33866,   2104,    9;
  9: 115975, 1194310, 3907353, 6239152, 5547365, 2830962, 796649, 107632, 4680, 10;
%F Formula
The row polynomials satisfy -(x-n*x^3)*P(n,x) = (2*n+(n^2+5*n-3)*x-4*x^2-(n^3+5*n^2+3*n-3)*x^3-(2*n^3+4*n^2-2*n-4)*x^4-(n^3-n)*x^5)*P(n-2,x) - (1+(n+4)*x+(2*n+4)*x^2-(n^2+2*n+2)*x^3-(3*n^2+7*n+5)*x^4-(3*n^2+5*n+2)*x^5-(n^2+n)*x^6)*d/dx P(n-2,x) - (2+(2*n+3)*x-2*x^2-(2*n^2+4*n+2)*x^3-2*n^2*x^4)*P(n-1,x) + (x+2*x^2-(n-1)*x^3-2*n*x^4-n*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + (n^2 + 5*n - 3)*x - 4*x^2 - (n^3 + 5*n^2 + 3*n - 3)*x^3 - (2*n^3 + 4*n^2 - 2*n - 4)*x^4 - (n^3 - n)*x^5)*P[n - 2, x] - (1 + (n + 4)*x + (2*n + 4)*x^2 - (n^2 + 2*n + 2)*x^3 - (3*n^2 + 7*n + 5)*x^4 - (3*n^2 + 5*n + 2)*x^5 - (n^2 + n)*x^6)*D[P[n - 2, x], x] - (2 + (2*n + 3)*x - 2*x^2 - (2*n^2 + 4*n + 2)*x^3 - 2*n^2*x^4)*P[n - 1, x] + (x + 2*x^2 - (n - 1)*x^3 - 2*n*x^4 - 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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + (n^2 + 5*n - 3)*x - 4*x^2 - (n^3 + 5*n^2 + 3*n - 3)*x^3 - (2*n^3 + 4*n^2 - 2*n - 4)*x^4 - (n^3 - n)*x^5)*v[i - 2] - (1 + (n + 4)*x + (2*n + 4)*x^2 - (n^2 + 2*n + 2)*x^3 - (3*n^2 + 7*n + 5)*x^4 - (3*n^2 + 5*n + 2)*x^5 - (n^2 + n)*x^6)*deriv(v[i - 2], x) - (2 + (2*n + 3)*x - 2*x^2 - (2*n^2 + 4*n + 2)*x^3 - 2*n^2*x^4)*v[i - 1] + (x + 2*x^2 - (n - 1)*x^3 - 2*n*x^4 - 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-9b9aa33d8b05d5b3
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-9b9aa33d8b05d5b3
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-9b9aa33d8b05d5b3

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

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.