Real-rooted atlas

A102365 → (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
\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-8\,x + 4\,x^{2} - n - 4\,n x - 28\,n x^{2} + n^{2} + 12\,n^{2} x + 53\,n^{2} x^{2} - 4\,n^{3} x - 38\,n^{3} x^{2} + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(3\,x - 7\,x^{2} + 5\,x^{3} + 4\,n x - 2\,n x^{2} - 32\,n x^{3} - 4\,n^{2} x + 53\,n^{2} x^{3} + 4\,n^{3} x^{2} - 30\,n^{3} x^{3} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(2\,x^{2} - 3\,x^{3} + x^{4} + 4\,n x^{2} + 10\,n x^{3} - 6\,n x^{4} - 4\,n^{2} x^{2} - 16\,n^{2} x^{3} + 9\,n^{2} x^{4} + 8\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(1 - 4\,x - n + 12\,n x - 12\,n^{2} x + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(x - 6\,n x - 4\,n x^{2} + 4\,n^{2} x + 8\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2145
3120930
417213516224
512321161592252011
6171678931033620753621057
7121724752012848067125637249251719

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-07729e5a1d997c68 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-ef3bb1c7737673b3

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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1,     4;
  3: 1,    20,       9;
  4: 1,    72,     135,       16;
  5: 1,   232,    1161,      592,       25;
  6: 1,   716,    7893,    10336,     2075,       36;
  7: 1,  2172,   47520,   128480,    67125,     6372,      49;
  8: 1,  6544,  266922,  1321760,  1407125,   360072,   17983,    64;
  9: 1, 19664, 1437678, 12099488, 22990375, 12065832, 1704073, 47936, 81;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = (n^2-n-(4*n^3-12*n^2+4*n+8)*x+(9*n^4-38*n^3+53*n^2-28*n+4)*x^2)*P(n-2,x) - ((4*n^2-4*n-3)*x-(4*n^3-2*n-7)*x^2-(4*n^4-30*n^3+53*n^2-32*n+5)*x^3)*d/dx P(n-2,x) - ((4*n^2-4*n-2)*x^2-(8*n^3-16*n^2+10*n-3)*x^3+(4*n^3-9*n^2+6*n-1)*x^4)*d^2/dx^2 P(n-2,x) - (n-1-(4*n^3-12*n^2+12*n-4)*x)*P(n-1,x) + ((4*n^2-6*n+1)*x-(4*n^3-8*n^2+4*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - n - (4*n^3 - 12*n^2 + 4*n + 8)*x + (9*n^4 - 38*n^3 + 53*n^2 - 28*n + 4)*x^2)*P[n - 2, x] - ((4*n^2 - 4*n - 3)*x - (4*n^3 - 2*n - 7)*x^2 - (4*n^4 - 30*n^3 + 53*n^2 - 32*n + 5)*x^3)*D[P[n - 2, x], x] - ((4*n^2 - 4*n - 2)*x^2 - (8*n^3 - 16*n^2 + 10*n - 3)*x^3 + (4*n^3 - 9*n^2 + 6*n - 1)*x^4)*D[P[n - 2, x], {x, 2}] - (n - 1 - (4*n^3 - 12*n^2 + 12*n - 4)*x)*P[n - 1, x] + ((4*n^2 - 6*n + 1)*x - (4*n^3 - 8*n^2 + 4*n)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 - n - (4*n^3 - 12*n^2 + 4*n + 8)*x + (9*n^4 - 38*n^3 + 53*n^2 - 28*n + 4)*x^2)*v[i - 2] - ((4*n^2 - 4*n - 3)*x - (4*n^3 - 2*n - 7)*x^2 - (4*n^4 - 30*n^3 + 53*n^2 - 32*n + 5)*x^3)*deriv(v[i - 2], x) - ((4*n^2 - 4*n - 2)*x^2 - (8*n^3 - 16*n^2 + 10*n - 3)*x^3 + (4*n^3 - 9*n^2 + 6*n - 1)*x^4)*deriv(deriv(v[i - 2], x), x) - (n - 1 - (4*n^3 - 12*n^2 + 12*n - 4)*x)*v[i - 1] + ((4*n^2 - 6*n + 1)*x - (4*n^3 - 8*n^2 + 4*n)*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); 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-872bb58370a3a8d0
  • 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-872bb58370a3a8d0
  • 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-872bb58370a3a8d0
  • 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-872bb58370a3a8d0

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.