Real-rooted atlas

A013622 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(3 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 4 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0459
112352572
236165250125576
3108675157516256254608
4324256581001275010000312536864
597293153712578750937505937515625294912
6291632805157950421875675000646875343750781252359296
787481129956378752055375413437553156254265625195312539062518874368
8262443827252478600935550022680000366187503937500027187500109375001953125150994944
97873212793959349425404595001148175002232562503012187502784375001687500006054687597656251207959552
1023619642318453444525016812562554675000012438562502019937500234140625018984375001025390625332031250488281259663676416
117085881387651512449497567660312524808781256465318750122790937501712390625017402343750125683593756123046875180664062524414062577309411328
122125764451724854428675002652284250108256500003180034687569163875000112767187500137826562500124716796875812109375003603515625097656250001220703125618475290624
1363772921461462751554464925101711902504573837125014952929062536649335937568412093750097731562500010632832031258672167968755141601562502094726562505249023437561035156254947802324992
1419131876470325285539412615038285895375188071065000677279728125184712653125038848296093756352551562500807642773437579180664062505878564453125319921875000012048339843752807617187503051757812539582418599936

Computed in your browser ?

Technical details
Node id
n-8ae41b2ed712d989 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-ee71b88eaf7760c9

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, 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 ?

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:     4,       5;
  1:    12,      35,      25;
  2:    36,     165,     250,      125;
  3:   108,     675,    1575,     1625,       625;
  4:   324,    2565,    8100,    12750,     10000,      3125;
  5:   972,    9315,   37125,    78750,     93750,     59375,     15625;
  6:  2916,   32805,  157950,   421875,    675000,    646875,    343750,     78125;
  7:  8748,  112995,  637875,  2055375,   4134375,   5315625,   4265625,   1953125,    390625;
  8: 26244,  382725, 2478600,  9355500,  22680000,  36618750,  39375000,  27187500,  10937500,  1953125;
  9: 78732, 1279395, 9349425, 40459500, 114817500, 223256250, 301218750, 278437500, 168750000, 60546875, 9765625;
%F Formula
The row polynomials satisfy P(n,x) = (3+5*x)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 5*x.
%t Mathematica
P[0, x_] := 4 + 5*x;
P[n_, x_] := P[n, x] = Expand[(3 + 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] = 4 + 5*x; for(n = 1, N, my(i = n + 1); v[i] = (3 + 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-19ee20f31e12280f

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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  4. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
  5. [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
  6. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.