Real-rooted atlas

A299989 → basis A013611

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(8 + 24\,x + 16\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0145
185611264240
264640236840963328102411520
3512665635328998401628161536007782416384552960
440966553645056017530884263936673382469140484456448163840026214426542080
532768622592524288025886720833945601842544642851471363093299202306867201127219203250585641943041274019840
6262144576716857409536342884352137232384038896926728037597184122662420481383176601611387535360665636044826172456966207569926710886461152952320
720971525242880060188262442131783682012636774469539463168179610583040353267351552533645492224619322540032547859988480362891182080174281719808573109698561154272460810737418242935341711360

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-f2d3e5cb5c029ac6 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-d3a321e8b3ce7458

Observed on the stored rows ?

  • Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, 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): yes

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,          4;
  1:         8,         56,         112,           64;
  2:        64,        640,        2368,         4096,          3328,           1024;
  3:       512,       6656,       35328,        99840,        162816,         153600,          77824,           16384;
  4:      4096,      65536,      450560,      1753088,       4263936,        6733824,        6914048,         4456448,         1638400,          262144;
  5:     32768,     622592,     5242880,     25886720,      83394560,      184254464,      285147136,       309329920,       230686720,       112721920,         32505856,          4194304;
  6:    262144,    5767168,    57409536,    342884352,    1372323840,     3889692672,     8037597184,     12266242048,     13831766016,     11387535360,       6656360448,       2617245696,        620756992,        67108864;
  7:   2097152,   52428800,   601882624,   4213178368,   20126367744,    69539463168,   179610583040,    353267351552,    533645492224,    619322540032,     547859988480,     362891182080,     174281719808,     57310969856,     11542724608,      1073741824;
  8:  16777216,  469762048,  6106906624,  48989470720,  271757344768,  1106759385088,  3427853664256,   8249424216064,  15621349703680,  23414349758464,   27784948744192,   25960929820672,   18869401944064,  10447507947520,   4256312590336,   1202590842880,   210453397504,    17179869184;
  9: 134217728, 4160749568, 60397977600, 545997717504, 3447516561408, 16160082886656, 58333172072448, 165972031832064, 377802637443072, 694217978413056, 1034165579415552, 1249155804561408, 1218576711155712, 951820587368448, 586701122568192, 278932356071424, 98646808854528, 24429773979648, 3779571220480, 274877906944;
%F Formula
The row polynomials satisfy P(n,x) = (8+24*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[(8 + 24*x + 16*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 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = (8 + 24*x + 16*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-6938cf52b8fdd4be
  • 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-6938cf52b8fdd4be
  • 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-6938cf52b8fdd4be

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.