Real-rooted atlas

A062196 → (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
\(\left(2\,n + 4\,n x + n^{2} - 10\,n^{2} x - 14\,n^{3} x + 8\,n^{3} x^{2} - 4\,n^{4} x + 12\,n^{4} x^{2} + 6\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(8\,x - 16\,x^{2} + 8\,x^{3} + 2\,n + 20\,n x - 66\,n x^{2} + 40\,n x^{3} + n^{2} + 2\,n^{2} x - 99\,n^{2} x^{2} + 84\,n^{2} x^{3} - 14\,n^{3} x - 78\,n^{3} x^{2} + 96\,n^{3} x^{3} - 4\,n^{4} x - 9\,n^{4} x^{2} + 69\,n^{4} x^{3} + 6\,n^{5} x^{2} + 24\,n^{5} x^{3} + n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(2\,x - 12\,x^{2} + 18\,x^{3} - 8\,x^{4} + 2\,n x - 32\,n x^{2} + 62\,n x^{3} - 32\,n x^{4} - 32\,n^{2} x^{2} + 84\,n^{2} x^{3} - 52\,n^{2} x^{4} - 8\,n^{3} x^{2} + 52\,n^{3} x^{3} - 44\,n^{3} x^{4} + 16\,n^{4} x^{3} - 16\,n^{4} x^{4} + 2\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111213
21325487
3160270160491
419681012803752562
5114018905600437575612762
611923780179202625012096137261611
7125268044704011025095256288122304290719

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-2e37cda1e7c758ee · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-3d8cfcfb4b7f185c

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: 1,  12;
  2: 1,  32,    54;
  3: 1,  60,   270,    160;
  4: 1,  96,   810,   1280,     375;
  5: 1, 140,  1890,   5600,    4375,     756;
  6: 1, 192,  3780,  17920,   26250,   12096,    1372;
  7: 1, 252,  6804,  47040,  110250,   95256,   28812,   2304;
  8: 1, 320, 11340, 107520,  367500,  508032,  288120,  61440,   3645;
  9: 1, 396, 17820, 221760, 1039500, 2095632, 1901592, 760320, 120285, 5500;
%F Formula
The row polynomials satisfy (n^2+2*n-(4*n^4+14*n^3+10*n^2-4*n)*x+(n^6+6*n^5+12*n^4+8*n^3)*x^2)*P(n,x) = (n^2+2*n-(4*n^4+14*n^3-2*n^2-20*n-8)*x+(n^6+6*n^5-9*n^4-78*n^3-99*n^2-66*n-16)*x^2+(3*n^6+24*n^5+69*n^4+96*n^3+84*n^2+40*n+8)*x^3)*P(n-1,x) + ((2*n+2)*x-(8*n^3+32*n^2+32*n+12)*x^2+(2*n^5+16*n^4+52*n^3+84*n^2+62*n+18)*x^3-(2*n^5+16*n^4+44*n^3+52*n^2+32*n+8)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 2*n - (4*n^4 + 14*n^3 - 2*n^2 - 20*n - 8)*x + (n^6 + 6*n^5 - 9*n^4 - 78*n^3 - 99*n^2 - 66*n - 16)*x^2 + (3*n^6 + 24*n^5 + 69*n^4 + 96*n^3 + 84*n^2 + 40*n + 8)*x^3)*P[n - 1, x] + ((2*n + 2)*x - (8*n^3 + 32*n^2 + 32*n + 12)*x^2 + (2*n^5 + 16*n^4 + 52*n^3 + 84*n^2 + 62*n + 18)*x^3 - (2*n^5 + 16*n^4 + 44*n^3 + 52*n^2 + 32*n + 8)*x^4)*D[P[n - 1, x], x])/(n^2 + 2*n - (4*n^4 + 14*n^3 + 10*n^2 - 4*n)*x + (n^6 + 6*n^5 + 12*n^4 + 8*n^3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 2*n - (4*n^4 + 14*n^3 - 2*n^2 - 20*n - 8)*x + (n^6 + 6*n^5 - 9*n^4 - 78*n^3 - 99*n^2 - 66*n - 16)*x^2 + (3*n^6 + 24*n^5 + 69*n^4 + 96*n^3 + 84*n^2 + 40*n + 8)*x^3)*v[i - 1] + ((2*n + 2)*x - (8*n^3 + 32*n^2 + 32*n + 12)*x^2 + (2*n^5 + 16*n^4 + 52*n^3 + 84*n^2 + 62*n + 18)*x^3 - (2*n^5 + 16*n^4 + 44*n^3 + 52*n^2 + 32*n + 8)*x^4)*deriv(v[i - 1], x))/(n^2 + 2*n - (4*n^4 + 14*n^3 + 10*n^2 - 4*n)*x + (n^6 + 6*n^5 + 12*n^4 + 8*n^3)*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)

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-a3706a3507b0ac5d
  • 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-a3706a3507b0ac5d
  • 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-a3706a3507b0ac5d
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-a3706a3507b0ac5d

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.