Real-rooted atlas

A106800 → (θ+1)²

nonnegative coefficients

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

Recurrence
\(\left(4 - 4\,n + n^{2}\right)\,P_{n}(x) = \left(-2 + 36\,x + 132\,x^{2} + 3\,n - 48\,n x - 284\,n x^{2} - n^{2} + 24\,n^{2} x + 221\,n^{2} x^{2} - 4\,n^{3} x - 74\,n^{3} x^{2} + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-x - 11\,x^{2} + 124\,x^{3} + 32\,n x^{2} - 244\,n x^{3} - 20\,n^{2} x^{2} + 167\,n^{2} x^{3} + 4\,n^{3} x^{2} - 46\,n^{3} x^{3} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(x^{3} + 20\,x^{4} - 36\,n x^{4} + 21\,n^{2} x^{4} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(6 - 32\,x - 7\,n + 48\,n x + 2\,n^{2} - 24\,n^{2} x + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(x + 16\,x^{2} - 32\,n x^{2} + 20\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
111
211
3145
4112922
51246316104
614022524025531
71605851440775362897
818412605600752522684916787

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-fe81f0230e45a5e7 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-dc8b66099215d7ce

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1;
   3: 1,   4;
   4: 1,  12,    9;
   5: 1,  24,   63,    16;
   6: 1,  40,  225,   240,     25;
   7: 1,  60,  585,  1440,    775,     36;
   8: 1,  84, 1260,  5600,   7525,   2268,     49;
   9: 1, 112, 2394, 16800,  42525,  34776,   6223,    64;
  10: 1, 144, 4158, 42336, 173775, 279720, 148225, 16320, 81;
%F Formula
The row polynomials satisfy (n^2-4*n+4)*P(n,x) = -(n^2-3*n+2+(4*n^3-24*n^2+48*n-36)*x-(9*n^4-74*n^3+221*n^2-284*n+132)*x^2)*P(n-2,x) - (x-(4*n^3-20*n^2+32*n-11)*x^2-(4*n^4-46*n^3+167*n^2-244*n+124)*x^3)*d/dx P(n-2,x) + (x^3-(4*n^3-21*n^2+36*n-20)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^2-7*n+6+(4*n^3-24*n^2+48*n-32)*x)*P(n-1,x) + (x-(4*n^3-20*n^2+32*n-16)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 3*n + 2 + (4*n^3 - 24*n^2 + 48*n - 36)*x - (9*n^4 - 74*n^3 + 221*n^2 - 284*n + 132)*x^2)*P[n - 2, x] - (x - (4*n^3 - 20*n^2 + 32*n - 11)*x^2 - (4*n^4 - 46*n^3 + 167*n^2 - 244*n + 124)*x^3)*D[P[n - 2, x], x] + (x^3 - (4*n^3 - 21*n^2 + 36*n - 20)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 - 7*n + 6 + (4*n^3 - 24*n^2 + 48*n - 32)*x)*P[n - 1, x] + (x - (4*n^3 - 20*n^2 + 32*n - 16)*x^2)*D[P[n - 1, x], x])/(n^2 - 4*n + 4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(n^2 - 3*n + 2 + (4*n^3 - 24*n^2 + 48*n - 36)*x - (9*n^4 - 74*n^3 + 221*n^2 - 284*n + 132)*x^2)*v[i - 2] - (x - (4*n^3 - 20*n^2 + 32*n - 11)*x^2 - (4*n^4 - 46*n^3 + 167*n^2 - 244*n + 124)*x^3)*deriv(v[i - 2], x) + (x^3 - (4*n^3 - 21*n^2 + 36*n - 20)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 7*n + 6 + (4*n^3 - 24*n^2 + 48*n - 32)*x)*v[i - 1] + (x - (4*n^3 - 20*n^2 + 32*n - 16)*x^2)*deriv(v[i - 1], x))/(n^2 - 4*n + 4)); 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, 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-39092568b301666d
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-39092568b301666d
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-39092568b301666d
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-39092568b301666d

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

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

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.