Real-rooted atlas

A038314 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(10 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1100 + 3840\,x + 4464\,x^{2} + 1728\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10Pn(1)
0110038404464172811132
11100051600907207084820736244904
21100006480001526400179712010575362488325387888
311000007800000230400003628800032140800151787522985984118533536
41100000091200000324000000639360000756864000537477120212004864358318082607737792
5110000000104400000043344000001028160000015240960000144571392008569774080290237644842998169657370231424
6110000000011760000000558720000001548288000002757888000003274629120002591834112001318610534403912833433651597803521262145091328
7110000000001308000000006998400000002218752000000461583360000065840947200006521389056000442881146880019736159846405211378155526191736422427767192009216

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-2a670a9a4a520101 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-5c5d2b4d4ee9e824

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:          1100,           3840,            4464,            1728;
  1:         11000,          51600,           90720,           70848,            20736;
  2:        110000,         648000,         1526400,         1797120,          1057536,           248832;
  3:       1100000,        7800000,        23040000,        36288000,         32140800,         15178752,          2985984;
  4:      11000000,       91200000,       324000000,       639360000,        756864000,        537477120,        212004864,         35831808;
  5:     110000000,     1044000000,      4334400000,     10281600000,      15240960000,      14457139200,       8569774080,       2902376448,        429981696;
  6:    1100000000,    11760000000,     55872000000,    154828800000,     275788800000,     327462912000,     259183411200,     131861053440,      39128334336,       5159780352;
  7:   11000000000,   130800000000,    699840000000,   2218752000000,    4615833600000,    6584094720000,    6521389056000,    4428811468800,    1973615984640,     521137815552,     61917364224;
  8:  110000000000,  1440000000000,   8568000000000,  30585600000000,   72783360000000,  121230950400000,  144223027200000,  122544783360000,   72881897472000,   28894769971200,   6872827428864,   743008370688;
  9: 1100000000000, 15720000000000, 102960000000000, 408672000000000, 1094860800000000, 2085709824000000, 2897001676800000, 2956124160000000, 2199356375040000, 1163530469376000, 415465513943040, 89904012853248, 8916100448256;
%F Formula
The row polynomials satisfy P(n,x) = (10+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1100 + 3840*x + 4464*x^2 + 1728*x^3.
%t Mathematica
P[0, x_] := 1100 + 3840*x + 4464*x^2 + 1728*x^3;
P[n_, x_] := P[n, x] = Expand[(10 + 12*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] = 1100 + 3840*x + 4464*x^2 + 1728*x^3; for(n = 1, N, my(i = n + 1); v[i] = (10 + 12*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-4dc01c187e0a54c1

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.