Real-rooted atlas

A173018 → (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
\(\left(-2 + n\right)\,P_{n}(x) = \left(-12\,x - 66\,x^{2} + 16\,n x + 109\,n x^{2} - 4\,n^{2} x - 56\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x + 5\,x^{2} - 62\,x^{3} - 2\,n x - 8\,n x^{2} + 91\,n x^{3} + 4\,n^{2} x^{2} - 38\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(x^{2} + 9\,x^{3} - 10\,x^{4} - n x^{2} - 12\,n x^{3} + 13\,n x^{4} + 4\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(-2 + 16\,x + n - 16\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x - 8\,x^{2} + 2\,n x + 12\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
111
211
3145
4116926
51449916160
61104594416251140
71228271848321425369240
8148010719386562977543204984000

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-11f5844271190e7f · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-3b4269c91cbc7ccc

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,   16,      9;
   5: 1,   44,     99,      16;
   6: 1,  104,    594,     416,      25;
   7: 1,  228,   2718,    4832,    1425,      36;
   8: 1,  480,  10719,   38656,   29775,    4320,     49;
   9: 1,  988,  38637,  249904,  390475,  154548,  12103,    64;
  10: 1, 2008, 131472, 1411744, 3904750, 3176424, 715792, 32128, 81;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = -((4*n^2-16*n+12)*x-(9*n^3-56*n^2+109*n-66)*x^2)*P(n-2,x) - ((2*n-2)*x-(4*n^2-8*n+5)*x^2-(4*n^3-38*n^2+91*n-62)*x^3)*d/dx P(n-2,x) - ((n-1)*x^2-(4*n^2-12*n+9)*x^3+(4*n^2-13*n+10)*x^4)*d^2/dx^2 P(n-2,x) + (n-2+(4*n^2-16*n+16)*x)*P(n-1,x) + ((2*n-3)*x-(4*n^2-12*n+8)*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[(-((4*n^2 - 16*n + 12)*x - (9*n^3 - 56*n^2 + 109*n - 66)*x^2)*P[n - 2, x] - ((2*n - 2)*x - (4*n^2 - 8*n + 5)*x^2 - (4*n^3 - 38*n^2 + 91*n - 62)*x^3)*D[P[n - 2, x], x] - ((n - 1)*x^2 - (4*n^2 - 12*n + 9)*x^3 + (4*n^2 - 13*n + 10)*x^4)*D[P[n - 2, x], {x, 2}] + (n - 2 + (4*n^2 - 16*n + 16)*x)*P[n - 1, x] + ((2*n - 3)*x - (4*n^2 - 12*n + 8)*x^2)*D[P[n - 1, x], x])/(n - 2)]];
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] = (-((4*n^2 - 16*n + 12)*x - (9*n^3 - 56*n^2 + 109*n - 66)*x^2)*v[i - 2] - ((2*n - 2)*x - (4*n^2 - 8*n + 5)*x^2 - (4*n^3 - 38*n^2 + 91*n - 62)*x^3)*deriv(v[i - 2], x) - ((n - 1)*x^2 - (4*n^2 - 12*n + 9)*x^3 + (4*n^2 - 13*n + 10)*x^4)*deriv(deriv(v[i - 2], x), x) + (n - 2 + (4*n^2 - 16*n + 16)*x)*v[i - 1] + ((2*n - 3)*x - (4*n^2 - 12*n + 8)*x^2)*deriv(v[i - 1], x))/(n - 2)); 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-55c66a5e0fb07f9a
  • 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-55c66a5e0fb07f9a
  • 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-55c66a5e0fb07f9a
  • 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-55c66a5e0fb07f9a

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.