Real-rooted atlas

A063967 → basis A007318

nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(2 + x\right)\,P_{n-2}(x) + \left(2 + x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2213
365112
416208145
5447243111171
612024419475141648
73287967904091161712457
88962528300819527441662019315
9244878721092085204081122622523135316
1066882414438256348722012276091882293261133893
1118272731681303681359609179841873130492739370291507627
1249920219584434560510288394672210884793442099538244563211924560
131363846536961422608185742416191889919844375431398613212251645513517296561
1437260819328644587616659259263954324419596223664283859923274847186678665538127663363
1510179845682112146070082291025624479256188377801075995046311051508200369570670248717768411104879772

Computed in your browser ?

Other recurrences

X ↦ x+1 substitution · proved equal for all n (proof step)

\(\left(-1 + n\right)\,P_{n}(x) = \left(6 + x - 2\,n\right)\,P_{n-1}(x) + \left(12 + 8\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-c63944481deaae45 · revision 2 · JSON
Definition source
derived:basis_map_A007318
derived:shift_x_plus_one (equal by e-ff5759bcc2b3a8b7)
Fingerprint
f6-bf438b133f65bda8

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:    2,     1;
   3:    6,     5,     1;
   4:   16,    20,     8,     1;
   5:   44,    72,    43,    11,     1;
   6:  120,   244,   194,    75,    14,    1;
   7:  328,   796,   790,   409,   116,   17,    1;
   8:  896,  2528,  3008,  1952,   744,  166,   20,   1;
   9: 2448,  7872, 10920,  8520,  4081, 1226,  225,  23,  1;
  10: 6688, 24144, 38256, 34872, 20122, 7609, 1882, 293, 26, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-2,x) + (2+x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 2, x] + (2 + x)*P[n - 1, 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] = 2 + x; for(n = 3, N, my(i = n); v[i] = (2 + x)*v[i - 2] + (2 + x)*v[i - 1]); 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-3a20d12bade77e7f
  • 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-3a20d12bade77e7f, e-4db5907b6eeb8d63
  • 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-3a20d12bade77e7f, e-4db5907b6eeb8d63
  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-4db5907b6eeb8d63

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. [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
  2. [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