Real-rooted atlas

A172431 → (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
\(\left(2\,n - 4\,n x + 8\,n^{2} - 8\,n^{2} x + 8\,n^{3} + 24\,n^{3} x + 8\,n^{3} x^{2} + 48\,n^{4} x + 32\,n^{4} x^{2} + 32\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-8\,x - 4\,x^{2} - 8\,x^{3} + 2\,n - 12\,n x - 88\,n x^{2} - 40\,n x^{3} + 8\,n^{2} + 56\,n^{2} x - 4\,n^{2} x^{2} - 40\,n^{2} x^{3} + 8\,n^{3} + 120\,n^{3} x + 344\,n^{3} x^{2} + 248\,n^{3} x^{3} + 48\,n^{4} x + 272\,n^{4} x^{2} + 320\,n^{4} x^{3} + 32\,n^{5} x^{2} + 96\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-x - 6\,x^{2} - 12\,x^{3} - 16\,x^{4} - 4\,n x - 40\,n x^{2} - 124\,n x^{3} - 112\,n x^{4} - 4\,n^{2} x - 64\,n^{2} x^{2} - 268\,n^{2} x^{3} - 304\,n^{2} x^{4} - 24\,n^{3} x^{2} - 160\,n^{3} x^{3} - 256\,n^{3} x^{4} - 16\,n^{4} x^{3} - 64\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1189
21162744
31249064179
4132189320125667
51403248968752162352
61484951920315020163437973
7156702352082509072411651226229

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-27a542f83a1a4ac0 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-ae96443b231c3935

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,  8;
  2: 1, 16,   27;
  3: 1, 24,   90,   64;
  4: 1, 32,  189,  320,   125;
  5: 1, 40,  324,  896,   875,   216;
  6: 1, 48,  495, 1920,  3150,  2016,   343;
  7: 1, 56,  702, 3520,  8250,  9072,  4116,   512;
  8: 1, 64,  945, 5824, 17875, 28512, 22638,  7680,   729;
  9: 1, 72, 1224, 8960, 34125, 72072, 84084, 50688, 13365, 1000;
%F Formula
The row polynomials satisfy (8*n^3+8*n^2+2*n+(48*n^4+24*n^3-8*n^2-4*n)*x+(32*n^5+32*n^4+8*n^3)*x^2)*P(n,x) = (8*n^3+8*n^2+2*n+(48*n^4+120*n^3+56*n^2-12*n-8)*x+(32*n^5+272*n^4+344*n^3-4*n^2-88*n-4)*x^2+(96*n^5+320*n^4+248*n^3-40*n^2-40*n-8)*x^3)*P(n-1,x) - ((4*n^2+4*n+1)*x+(24*n^3+64*n^2+40*n+6)*x^2+(16*n^4+160*n^3+268*n^2+124*n+12)*x^3+(64*n^4+256*n^3+304*n^2+112*n+16)*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[((8*n^3 + 8*n^2 + 2*n + (48*n^4 + 120*n^3 + 56*n^2 - 12*n - 8)*x + (32*n^5 + 272*n^4 + 344*n^3 - 4*n^2 - 88*n - 4)*x^2 + (96*n^5 + 320*n^4 + 248*n^3 - 40*n^2 - 40*n - 8)*x^3)*P[n - 1, x] - ((4*n^2 + 4*n + 1)*x + (24*n^3 + 64*n^2 + 40*n + 6)*x^2 + (16*n^4 + 160*n^3 + 268*n^2 + 124*n + 12)*x^3 + (64*n^4 + 256*n^3 + 304*n^2 + 112*n + 16)*x^4)*D[P[n - 1, x], x])/(8*n^3 + 8*n^2 + 2*n + (48*n^4 + 24*n^3 - 8*n^2 - 4*n)*x + (32*n^5 + 32*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] = ((8*n^3 + 8*n^2 + 2*n + (48*n^4 + 120*n^3 + 56*n^2 - 12*n - 8)*x + (32*n^5 + 272*n^4 + 344*n^3 - 4*n^2 - 88*n - 4)*x^2 + (96*n^5 + 320*n^4 + 248*n^3 - 40*n^2 - 40*n - 8)*x^3)*v[i - 1] - ((4*n^2 + 4*n + 1)*x + (24*n^3 + 64*n^2 + 40*n + 6)*x^2 + (16*n^4 + 160*n^3 + 268*n^2 + 124*n + 12)*x^3 + (64*n^4 + 256*n^3 + 304*n^2 + 112*n + 16)*x^4)*deriv(v[i - 1], x))/(8*n^3 + 8*n^2 + 2*n + (48*n^4 + 24*n^3 - 8*n^2 - 4*n)*x + (32*n^5 + 32*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-f7d0ac20788e6eb3
  • 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-f7d0ac20788e6eb3
  • 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-f7d0ac20788e6eb3
  • 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-f7d0ac20788e6eb3

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023