Real-rooted atlas

A013612 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(1 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2 + 25\,x + 100\,x^{2} + 125\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17Pn(1)
0225100125252
12352256256251512
22454001750375031259072
3255625375012500218751562554432
42659006875312508437512500078125326592
5275122511375656252406255468757031253906251959552
6285160017500122500568750175000034375003906250195312511757312
729520252550021000011812504593750121875002109375021484375976562570543872
82105250035625337500223125010500000351562508203125012695312511718750048828125423263232
92115302548125515625391875021656250876562502578125005371093757519531256347656252441406252539579392
1021253600632507562506496875412500001959375006960937501826171875343750000043945312503417968750122070312515237476352
112135422581250107250010278125737343754021875001675781250530664062512568359375215820312502539062500018310546875610351562591424858112
122145490010237514787501564062512512500077085937536867187501368554687539101562500844238281251333007812501452636718759765625000030517578125548549148672
132155562512687519906252303437520332812513964843757541015625321191406251075292968752799316406255554199218758117675781258239746093755187988281251525878906253291294892032
1421656400155000262500032987500318500000241312500014523437500698242187502681250000008175781250001955078125000358886718750048828125000004638671875000274658203125076293945312519747769352192

Computed in your browser ?

Technical details
Node id
n-6dca6e79e178d866 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-ec0bf7acaf45f506

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: 2,  25,  100,   125;
  1: 2,  35,  225,   625,    625;
  2: 2,  45,  400,  1750,   3750,    3125;
  3: 2,  55,  625,  3750,  12500,   21875,    15625;
  4: 2,  65,  900,  6875,  31250,   84375,   125000,    78125;
  5: 2,  75, 1225, 11375,  65625,  240625,   546875,   703125,    390625;
  6: 2,  85, 1600, 17500, 122500,  568750,  1750000,  3437500,   3906250,   1953125;
  7: 2,  95, 2025, 25500, 210000, 1181250,  4593750, 12187500,  21093750,  21484375,   9765625;
  8: 2, 105, 2500, 35625, 337500, 2231250, 10500000, 35156250,  82031250, 126953125, 117187500,  48828125;
  9: 2, 115, 3025, 48125, 515625, 3918750, 21656250, 87656250, 257812500, 537109375, 751953125, 634765625, 244140625;
%F Formula
The row polynomials satisfy P(n,x) = (1+5*x)*P(n-1,x) for n >= 1, with P(0,x) = 2 + 25*x + 100*x^2 + 125*x^3.
%t Mathematica
P[0, x_] := 2 + 25*x + 100*x^2 + 125*x^3;
P[n_, x_] := P[n, x] = Expand[(1 + 5*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] = 2 + 25*x + 100*x^2 + 125*x^3; for(n = 1, N, my(i = n + 1); v[i] = (1 + 5*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-0f5b357fc0453aeb

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.