Real-rooted atlas

A013620 → D+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
\(\left(-1 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(4 + 12\,x + 9\,x^{2} + 6\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1538
21630955
34414413527350
4112528864540812125
5272168039604320202524312500
66404896151202376019440729072971875
71472134405140810584012474081648255152187406250
83328353281612804112646350405987523265928748065612265625
9742489856476928145152027760323429216269438412597122952451968312500000
1016384222720134784047692801088640016656192171460801154736047239209841505904968359375
11358405406723674880148262403934656071850240916090568083152047632860173210403247695177147371093750
1277824129024097320964409856013343616028329523243110144047113228836374184019053144062355744106288205314412001953125
131679363035136251596801265172484299609601040802048184141900824018508802296769904157621464074307261622107945634543665159432310742187500
1436044870533126373785635223552013284311043611672064728561433611048514048126097171201071825955266201014882837186352773778096111602610478296957373046875

Computed in your browser ?

Technical details
Node id
n-2eb61489a761e52c · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bff12ef9596281bf

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:    5,     3;
  2:   16,    30,      9;
  3:   44,   144,    135,      27;
  4:  112,   528,    864,     540,      81;
  5:  272,  1680,   3960,    4320,    2025,     243;
  6:  640,  4896,  15120,   23760,   19440,    7290,     729;
  7: 1472, 13440,  51408,  105840,  124740,   81648,   25515,    2187;
  8: 3328, 35328, 161280,  411264,  635040,  598752,  326592,   87480,   6561;
  9: 7424, 89856, 476928, 1451520, 2776032, 3429216, 2694384, 1259712, 295245, 19683;
%F Formula
The row polynomials satisfy (3*n-1+3*x)*P(n,x) = (6*n+4+(9*n+12)*x+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[((6*n + 4 + (9*n + 12)*x + 9*x^2)*P[n - 1, x])/(3*n - 1 + 3*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] = ((6*n + 4 + (9*n + 12)*x + 9*x^2)*v[i - 1])/(3*n - 1 + 3*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-1ea66849eac85f55
  • 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-1ea66849eac85f55
  • 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-1ea66849eac85f55
  • 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-1ea66849eac85f55

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. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.