Real-rooted atlas

A038336 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(12 + 10\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1872 + 4560\,x + 3700\,x^{2} + 1000\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10Pn(1)
0187245603700100011132
12246473440900004900010000244904
22695681105920181440014880006100001000005387888
332348161596672032832000360000002220000073000001000000118533536
43881779222394880055365120076032000062640000030960000085000000100000002607737792
54658135043075563520888330240014660352000151200000009979200000411600000097000000010000000057370231424
6558976204841564897280137355264000264757248000328043520000270950400000149184000000528000000001090000000010000000001262145091328
7670771445765546763878402063912140800455063961600065840947200006531840000000449971200000021254400000006588000000001210000000001000000000027767192009216

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-bc204169a6a7086c · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-3f20e2f2cbb7dd50

Observed on the stored rows ?

  • Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 3)
  • 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 ?

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:          1872,           4560,            3700,             1000;
  1:         22464,          73440,           90000,            49000,            10000;
  2:        269568,        1105920,         1814400,          1488000,           610000,           100000;
  3:       3234816,       15966720,        32832000,         36000000,         22200000,          7300000,          1000000;
  4:      38817792,      223948800,       553651200,        760320000,        626400000,        309600000,         85000000,         10000000;
  5:     465813504,     3075563520,      8883302400,      14660352000,      15120000000,       9979200000,       4116000000,        970000000,        100000000;
  6:    5589762048,    41564897280,    137355264000,     264757248000,     328043520000,     270950400000,     149184000000,      52800000000,      10900000000,      1000000000;
  7:   67077144576,   554676387840,   2063912140800,    4550639616000,    6584094720000,    6531840000000,    4499712000000,    2125440000000,     658800000000,    121000000000,    10000000000;
  8:  804925734912,  7326888099840,  30313709568000,   75246796800000,  124515532800000,  144223027200000,  119314944000000,   70502400000000,   29160000000000,   8040000000000,  1330000000000,   100000000000;
  9: 9659108818944, 95971914547200, 437033395814400, 1206098657280000, 2246654361600000, 2975831654400000, 2874009600000000, 2039178240000000, 1054944000000000, 388080000000000, 96360000000000, 14500000000000, 1000000000000;
%F Formula
The row polynomials satisfy P(n,x) = (12+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 1872 + 4560*x + 3700*x^2 + 1000*x^3.
%t Mathematica
P[0, x_] := 1872 + 4560*x + 3700*x^2 + 1000*x^3;
P[n_, x_] := P[n, x] = Expand[(12 + 10*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] = 1872 + 4560*x + 3700*x^2 + 1000*x^3; for(n = 1, N, my(i = n + 1); v[i] = (12 + 10*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, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-0e5e881cd5b1385e

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.

References

  1. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
  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. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.