Real-rooted atlas

A013620 → θ+1

nonnegative coefficients only rational roots

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

Recurrence
\(\left(2 + 3\,n x\right)\,P_{n}(x) = \left(4 + 12\,x + 9\,x^{2} + 6\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1268
24242755
3872162108350
4161926488644052125
532480216043204050145812500
66411526480172802430017496510371875
7128268818144604801134001224727144217496406250
8256614448384193536453600653184571536279936590492265625
9512138241244165806081632960293932834292162519424106288219683012500000
101024307203110401658880544320011757312171460801679616010628820393660064953968359375
112048675847603204561920171072004311014475442752923788807794468043302600142898582125764371093750
124096147456182476812165120513216001478062083017710084434186244676680803464208001714782965101833669087332001953125
138192319488431308831629312148262400480370176112086374419214807042431874016225173520014861452326632383681796270582232052210742187500
14163846881281006387280510976415134720149448499239230231047685922816113487454081260971712010403016624619022476825147788126249746167174453557373046875

Computed in your browser ?

Technical details
Node id
n-eb2d732692093d3f · revision 1 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-311883d116e971c2

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): yes

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:   2,     6;
  2:   4,    24,     27;
  3:   8,    72,    162,    108;
  4:  16,   192,    648,    864,     405;
  5:  32,   480,   2160,   4320,    4050,    1458;
  6:  64,  1152,   6480,  17280,   24300,   17496,    5103;
  7: 128,  2688,  18144,  60480,  113400,  122472,   71442,   17496;
  8: 256,  6144,  48384, 193536,  453600,  653184,  571536,  279936,   59049;
  9: 512, 13824, 124416, 580608, 1632960, 2939328, 3429216, 2519424, 1062882, 196830;
%F Formula
The row polynomials satisfy (2+3*n*x)*P(n,x) = (4+(6*n+12)*x+(9*n+9)*x^2)*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[((4 + (6*n + 12)*x + (9*n + 9)*x^2)*P[n - 1, x])/(2 + 3*n*x)]];
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] = ((4 + (6*n + 12)*x + (9*n + 9)*x^2)*v[i - 1])/(2 + 3*n*x)); 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-7046ce3eaa519f49
  • 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-7046ce3eaa519f49
  • 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-7046ce3eaa519f49
  • 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-7046ce3eaa519f49

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.