Real-rooted atlas

A007318 → (θ+1)² → x ↦ x+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(8 + 12\,x + 6\,x^{2} + x^{3}\right)\,P_{n-3}(x) + \left(-12 - 12\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 4\,x\)
\(P_{2}(x) = 18 + 26\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1549
21826953
3561147516261
4160416396164251161
5432136016801020305364833
6112041286240496021905104919197
728161187221168207201204041587916473629

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-39becb5dff639ff2 · revision 1 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-2f7d4833c4308026

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:     5,     4;
  2:    18,    26,      9;
  3:    56,   114,     75,     16;
  4:   160,   416,    396,    164,     25;
  5:   432,  1360,   1680,   1020,    305,     36;
  6:  1120,  4128,   6240,   4960,   2190,    510,    49;
  7:  2816, 11872,  21168,  20720,  12040,   4158,   791,    64;
  8:  6912, 32768,  67200,  77952,  56000,  25536,  7224,  1160,   81;
  9: 16640, 87552, 202752, 271488, 231840, 131040, 49056, 11736, 1629, 100;
%F Formula
The row polynomials satisfy P(n,x) = (8+12*x+6*x^2+x^3)*P(n-3,x) - (12+12*x+3*x^2)*P(n-2,x) + (6+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 4*x, P(2,x) = 18 + 26*x + 9*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 4*x;
P[2, x_] := 18 + 26*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 12*x + 6*x^2 + x^3)*P[n - 3, x] - (12 + 12*x + 3*x^2)*P[n - 2, x] + (6 + 3*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] = 1; v[2] = 5 + 4*x; v[3] = 18 + 26*x + 9*x^2; for(n = 3, N, my(i = n + 1); v[i] = (8 + 12*x + 6*x^2 + x^3)*v[i - 3] - (12 + 12*x + 3*x^2)*v[i - 2] + (6 + 3*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, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-c354d1860a6cfc2f
  • 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-c354d1860a6cfc2f
  • 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-c354d1860a6cfc2f

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.