Real-rooted atlas

A128099 → (θ+1)²

nonnegative coefficients

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

Recurrence
\(\left(n - 4\,n x + n^{2} + 6\,n^{3} x + 4\,n^{3} x^{2} + 4\,n^{4} x^{2}\right)\,P_{n}(x) = \left(-16\,x - 16\,x^{2} - 64\,x^{3} + n - 4\,n x - 56\,n x^{2} - 64\,n x^{3} + n^{2} + 16\,n^{2} x + 32\,n^{2} x^{2} + 48\,n^{2} x^{3} + 6\,n^{3} x + 44\,n^{3} x^{2} + 64\,n^{3} x^{3} + 4\,n^{4} x^{2} + 16\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(-x - 12\,x^{2} - 48\,x^{3} - 128\,x^{4} - n x - 20\,n x^{2} - 128\,n x^{3} - 256\,n x^{4} - 6\,n^{2} x^{2} - 68\,n^{2} x^{3} - 160\,n^{2} x^{4} - 4\,n^{3} x^{3} - 32\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2189
311617
41243661
5132108141
6140216128385
7148360512921

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-00cae35e2dcc4f7b · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-ad3c4a2e4bd3c03a

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2: 1,  8;
  3: 1, 16;
  4: 1, 24,  36;
  5: 1, 32, 108;
  6: 1, 40, 216,  128;
  7: 1, 48, 360,  512;
  8: 1, 56, 540, 1280,  400;
  9: 1, 64, 756, 2560, 2000;
%F Formula
The row polynomials satisfy (n^2+n+(6*n^3-4*n)*x+(4*n^4+4*n^3)*x^2)*P(n,x) = (n^2+n+(6*n^3+16*n^2-4*n-16)*x+(4*n^4+44*n^3+32*n^2-56*n-16)*x^2+(16*n^4+64*n^3+48*n^2-64*n-64)*x^3)*P(n-1,x) - ((n+1)*x+(6*n^2+20*n+12)*x^2+(4*n^3+68*n^2+128*n+48)*x^3+(32*n^3+160*n^2+256*n+128)*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 + n + (6*n^3 + 16*n^2 - 4*n - 16)*x + (4*n^4 + 44*n^3 + 32*n^2 - 56*n - 16)*x^2 + (16*n^4 + 64*n^3 + 48*n^2 - 64*n - 64)*x^3)*P[n - 1, x] - ((n + 1)*x + (6*n^2 + 20*n + 12)*x^2 + (4*n^3 + 68*n^2 + 128*n + 48)*x^3 + (32*n^3 + 160*n^2 + 256*n + 128)*x^4)*D[P[n - 1, x], x])/(n^2 + n + (6*n^3 - 4*n)*x + (4*n^4 + 4*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 + n + (6*n^3 + 16*n^2 - 4*n - 16)*x + (4*n^4 + 44*n^3 + 32*n^2 - 56*n - 16)*x^2 + (16*n^4 + 64*n^3 + 48*n^2 - 64*n - 64)*x^3)*v[i - 1] - ((n + 1)*x + (6*n^2 + 20*n + 12)*x^2 + (4*n^3 + 68*n^2 + 128*n + 48)*x^3 + (32*n^3 + 160*n^2 + 256*n + 128)*x^4)*deriv(v[i - 1], x))/(n^2 + n + (6*n^3 - 4*n)*x + (4*n^4 + 4*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-97ccb3d326f9aac4
  • 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-97ccb3d326f9aac4
  • 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-97ccb3d326f9aac4

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.