Real-rooted atlas

A008277 → (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
\(P_{n}(x) = \left(-4\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 4\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} + 4\,x^{3}\right)\,P_{n-2}''(x) + \left(1 + 4\,x\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2145
3112922
4128541699
516022516025471
611248101040375362386
712522709560035007564912867
8150886942721626250957613726473681

Show all 15 stored rows

Computed in your browser ?

Other recurrences

Image under θ+1 · proved equal for all n (proof step)

\(\left(4\right)\,P_{n}(x) = \left(6\,x - 18\,x^{2} + 4\,x^{3}\right)\,P_{n-3}(x) + \left(3\,x - 9\,x^{2}\right)\,P_{n-3}'(x) + \left(x^{2} - 4\,x^{3}\right)\,P_{n-3}''(x) + \left(-6 + 12\,x - 12\,x^{2}\right)\,P_{n-2}(x) + \left(-2 + 4\,x\right)\,P_{n-2}'(x) + \left(-x + 4\,x^{2}\right)\,P_{n-2}''(x) + \left(6 + 12\,x\right)\,P_{n-1}(x) + \left(1\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 4\,x\)
\(P_{3}(x) = 1 + 12\,x + 9\,x^{2}\)
Technical details
Node id
n-0abf36e1d724d74f · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
derived:theta_plus_one (equal by e-6385ba87ef72ed57)
Fingerprint
f6-915ca4b22aed877c

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 -1)
  • 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 ?

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,    4;
   3: 1,   12,     9;
   4: 1,   28,    54,     16;
   5: 1,   60,   225,    160,      25;
   6: 1,  124,   810,   1040,     375,     36;
   7: 1,  252,  2709,   5600,    3500,    756,     49;
   8: 1,  508,  8694,  27216,   26250,   9576,   1372,    64;
   9: 1, 1020, 27225, 124320,  173775,  95256,  22638,  2304,   81;
  10: 1, 2044, 83970, 545680, 1063125, 821772, 288120, 48000, 3645, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(4*x-9*x^2)*P(n-2,x) - (2*x-4*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (1+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(4*x - 9*x^2)*P[n - 2, x] - (2*x - 4*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (1 + 4*x)*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], x]];
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 + 4*x; for(n = 3, N, my(i = n); v[i] = -(4*x - 9*x^2)*v[i - 2] - (2*x - 4*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (1 + 4*x)*v[i - 1] + (2*x - 4*x^2)*deriv(v[i - 1], x)); 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-72cd2be7747bf330, e-b4b600b457a94a2b
  • 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-72cd2be7747bf330, e-b4b600b457a94a2b
  • 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-72cd2be7747bf330, e-b4b600b457a94a2b
  • 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-72cd2be7747bf330

Proof chains

3 incoming proof steps. 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.