Real-rooted atlas

A038338 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(12 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1872 + 5472\,x + 5328\,x^{2} + 1728\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10Pn(1)
0187254725328172814400
122464881281296008467220736345600
226956813271042612736257126412648962488328294400
3323481619160064472780806220800046033920181647362985984199065600
43881779226873856079725772813138329601298903040770383872253808640358318084777574400
5465813504369067622412791955456253330882563135283200024831442944122903101443475685376429981696114661785600
65589762048498778767361977915801604575005245446802310430726742112993284454610370561891919462404686800486451597803522751882854400
76707714457666561166540829720334827527863505256448136527788113921625330810880013436068036608761583579955228327194132486243334225926191736422466045188505600

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-7eae85e3777acca0 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-ca6ef6daeaeecdff

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,            5472,            5328,             1728;
  1:         22464,           88128,          129600,            84672,            20736;
  2:        269568,         1327104,         2612736,          2571264,          1264896,           248832;
  3:       3234816,        19160064,        47278080,         62208000,         46033920,         18164736,          2985984;
  4:      38817792,       268738560,       797257728,       1313832960,       1298903040,        770383872,        253808640,         35831808;
  5:     465813504,      3690676224,     12791955456,      25333088256,      31352832000,      24831442944,      12290310144,       3475685376,        429981696;
  6:    5589762048,     49877876736,    197791580160,     457500524544,     680231043072,     674211299328,     445461037056,     189191946240,      46868004864,       5159780352;
  7:   67077144576,    665611665408,   2972033482752,    7863505256448,   13652778811392,   16253308108800,   13436068036608,    7615835799552,    2832719413248,     624333422592,     61917364224;
  8:  804925734912,   8792265719808,  43651741777920,  130026464870400,  258195408814080,  358873043042304,  356272513744896,  252622846033920,  125382662553600,   41484634030080,   8235009441792,    743008370688;
  9: 9659108818944, 115166297456640, 629328089972736, 2084138479779840, 4658662484213760, 7404821422276608, 8581746681446400, 7306744317345792, 4536066103050240, 2002407559004160, 596635721662464, 107736213749760, 8916100448256;
%F Formula
The row polynomials satisfy P(n,x) = (12+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1872 + 5472*x + 5328*x^2 + 1728*x^3.
%t Mathematica
P[0, x_] := 1872 + 5472*x + 5328*x^2 + 1728*x^3;
P[n_, x_] := P[n, x] = Expand[(12 + 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] = 1872 + 5472*x + 5328*x^2 + 1728*x^3; for(n = 1, N, my(i = n + 1); v[i] = (12 + 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-328ff92d3afa5881

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.