Real-rooted atlas

A059473 → x ↦ x+1

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(2 + 2\,x\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
21818440
380120568264
4356712508152161744
515843960380817523843211520
6704821144255921594453769286476096
73136010976016019212608057664152962176128502656

Show all 15 stored rows

Computed in your browser ?

Other recurrences

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

\(\left(n\right)\,P_{n}(x) = \left(8 + 4\,x - 7\,n - 2\,n x + 3\,n^{2}\right)\,P_{n-1}(x) + \left(30 + 36\,x + 8\,x^{2} - 12\,n - 12\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(12 + 24\,x + 14\,x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-43836771acfe9d89 · revision 2 · JSON
Definition source
derived:basis_map_A007318
derived:shift_x_plus_one (equal by e-7c5de25572de6f9f)
Fingerprint
f6-ab8a6f908e9a31d4

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 ?

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:      4,       2;
  2:     18,      18,       4;
  3:     80,     120,      56,       8;
  4:    356,     712,     508,     152,      16;
  5:   1584,    3960,    3808,    1752,     384,      32;
  6:   7048,   21144,   25592,   15944,    5376,     928,     64;
  7:  31360,  109760,  160192,  126080,   57664,   15296,   2176,    128;
  8: 139536,  558144,  953760,  907776,  525456,  189120,  41280,   4992,   256;
  9: 620864, 2793888, 5471232, 6111168, 4284864, 1953312, 578304, 107136, 11264, 512;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*x)*P(n-2,x) + (4+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] + (4 + 2*x)*P[n - 1, x]];
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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 2] + (4 + 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 - 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-d50592d294a2f5cf
  • 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-c090df23694a462d, e-d50592d294a2f5cf
  • 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-c090df23694a462d, e-d50592d294a2f5cf
  • 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-c090df23694a462d, e-d50592d294a2f5cf
  • Real-rooted. Every row P_n, n >= 0, 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-c090df23694a462d

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

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

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