Real-rooted atlas

A034870 → x ↦ x+1

nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14419
21632248181
36419224016060121729
425610241792179211204481121616561
5102451201152015360134408064336096018020159049
640962457667584112640126720101376591362534479201760264241531441
71638411468837273674547210250241025024768768439296192192640641601629123642814782969

Show all 15 stored rows

Generated b-files: A001019 (row sums), A100062 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-8c0973a753fcbe28 · revision 2 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-90334921cba1a1bb

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): yes

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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,       4,        1;
  2:     16,      32,       24,        8,        1;
  3:     64,     192,      240,      160,       60,       12,        1;
  4:    256,    1024,     1792,     1792,     1120,      448,      112,       16,        1;
  5:   1024,    5120,    11520,    15360,    13440,     8064,     3360,      960,      180,       20,        1;
  6:   4096,   24576,    67584,   112640,   126720,   101376,    59136,    25344,     7920,     1760,      264,      24,       1;
  7:  16384,  114688,   372736,   745472,  1025024,  1025024,   768768,   439296,   192192,    64064,    16016,    2912,     364,     28,     1;
  8:  65536,  524288,  1966080,  4587520,  7454720,  8945664,  8200192,  5857280,  3294720,  1464320,   512512,  139776,   29120,   4480,   480,   32,   1;
  9: 262144, 2359296, 10027008, 26738688, 50135040, 70189056, 76038144, 65175552, 44808192, 24893440, 11202048, 4073472, 1188096, 274176, 48960, 6528, 612, 36, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+4*x+x^2)*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[(4 + 4*x + x^2)*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; for(n = 1, N, my(i = n + 1); v[i] = (4 + 4*x + x^2)*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-f2e5e2864fad238b
  • 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-cfe8f3b6680af0c3, e-f2e5e2864fad238b
  • 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-cfe8f3b6680af0c3, e-f2e5e2864fad238b
  • 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-cfe8f3b6680af0c3

Proof chains

2 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 (5)

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