Real-rooted atlas

A013610 → (θ+1)²

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence
\(\left(1 - 3\,x + 9\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1 + 9\,x + 27\,x^{2} + 27\,x^{3} + 9\,n x + 45\,n x^{2} + 54\,n x^{3} + 9\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111213
212481106
3136243432712
4148486172820254288
5160810432010125874824064
617212158640303755248835721128512
718417011512070875183708250047139968661504
8196226824192141750489888100018811197445314413309568
911082916362882551501102248300056450388484782969196830016187392
10112036455184042525022044967501410167961602391484519683000714492977725696
11113244557128066825040415761650310246189440876877651082565007859421925509168367525888
1211445346950401002375692841633006204110854656263063295433026000471565314306110016898135291715470336
13115663181235521447875112586766129723624018508868396456714073345002043449694198971510411675758773124873087918845952
1411687371157248202702517513496107270163480370176159591732339405366007152073929928533715281730311394374822312107616802536205232128

Computed in your browser ?

Technical details
Node id
n-70aa978122334d17 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-10c18c8945c6e24a

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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): no

OEIS matches ?

No 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: 1;
  1: 1,  12;
  2: 1,  24,   81;
  3: 1,  36,  243,   432;
  4: 1,  48,  486,  1728,   2025;
  5: 1,  60,  810,  4320,  10125,    8748;
  6: 1,  72, 1215,  8640,  30375,   52488,   35721;
  7: 1,  84, 1701, 15120,  70875,  183708,  250047,  139968;
  8: 1,  96, 2268, 24192, 141750,  489888, 1000188, 1119744,  531441;
  9: 1, 108, 2916, 36288, 255150, 1102248, 3000564, 5038848, 4782969, 1968300;
%F Formula
The row polynomials satisfy (1+(9*n-3)*x+9*n^2*x^2)*P(n,x) = (1+(9*n+9)*x+(9*n^2+45*n+27)*x^2+(27*n^2+54*n+27)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (9*n + 9)*x + (9*n^2 + 45*n + 27)*x^2 + (27*n^2 + 54*n + 27)*x^3)*P[n - 1, x])/(1 + (9*n - 3)*x + 9*n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (9*n + 9)*x + (9*n^2 + 45*n + 27)*x^2 + (27*n^2 + 54*n + 27)*x^3)*v[i - 1])/(1 + (9*n - 3)*x + 9*n^2*x^2)); 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, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-30efefd246e11889
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-30efefd246e11889
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-30efefd246e11889
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-30efefd246e11889

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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  4. [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
  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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.