Real-rooted atlas

A106800 → reverse (shift +1) → θ+1

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(2\,x - x^{2}\right)\,P_{n-2}(x) + \left(x - x^{2}\right)\,P_{n-2}'(x) + \left(-1 + 2\,x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1022
20235
3029415
4022124552
50245100506203
60293360325907877
7021891204175084014784140

Show all 15 stored rows

Generated b-files: A069138 ?

Computed in your browser ?

Technical details
Node id
n-6d092678b88e72ae · revision 1 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8452b60480f5fd7f

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 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 0, 2;
  2: 0, 2,   3;
  3: 0, 2,   9,     4;
  4: 0, 2,  21,    24,     5;
  5: 0, 2,  45,   100,    50,     6;
  6: 0, 2,  93,   360,   325,    90,     7;
  7: 0, 2, 189,  1204,  1750,   840,   147,    8;
  8: 0, 2, 381,  3864,  8505,  6300,  1862,  224,   9;
  9: 0, 2, 765, 12100, 38850, 41706, 18522, 3696, 324, 10;
%F Formula
The row polynomials satisfy P(n,x) = (2*x-x^2)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) - (1-2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[(2*x - x^2)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] - (1 - 2*x)*P[n - 1, x] + x*D[P[n - 1, x], 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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2*x - x^2)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) - (1 - 2*x)*v[i - 1] + x*deriv(v[i - 1], 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.

A069138: triangle, offset 1, from the row P1, low zeros dropped
%e Example
Triangle begins:
   1: 2;
   2: 2,    3;
   3: 2,    9,     4;
   4: 2,   21,    24,      5;
   5: 2,   45,   100,     50,      6;
   6: 2,   93,   360,    325,     90,      7;
   7: 2,  189,  1204,   1750,    840,    147,     8;
   8: 2,  381,  3864,   8505,   6300,   1862,   224,    9;
   9: 2,  765, 12100,  38850,  41706,  18522,  3696,  324,  10;
  10: 2, 1533, 37320, 170525, 255150, 159789, 47040, 6750, 450, 11;
%F Formula
Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = (2*x-x^2)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) - (1-2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2*x, P(2,x) = 2*x + 3*x^2.
%t Mathematica
P[1, x_] := 2*x;
P[2, x_] := 2*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[(2*x - x^2)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] - (1 - 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2*x; v[2] = 2*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = (2*x - x^2)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) - (1 - 2*x)*v[i - 1] + x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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-315f41466485bb90
  • 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-315f41466485bb90
  • 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-315f41466485bb90

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

Same fingerprint (2)

Equal up to content, sign, reversal and a small index shift.

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.