Real-rooted atlas

A159854 → Veronese 2 (j=1)

nonnegative coefficients

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

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
00
1011
2022
30314
40448
50510116
60620632
7073521164
80856568128
90984126361256
1001012025212010512
110111654623305511024
12012220792792220122048
13013286128717167157814096
14014364200234322002364148192

Generated b-files: A000079 (row sums), A034867, A109447 (reversed triangle), A131577 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-1f922d5ae3a4fc22 · revision 1 · JSON
Definition source
derived:veronese_section_2_1
Fingerprint
f6-937db0732045a112

Observed on the stored rows ?

  • Degrees: -1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half affine: slope 1, intercept 1)
  • 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: 0;
  1: 0, 1;
  2: 0, 2;
  3: 0, 3,  1;
  4: 0, 4,  4;
  5: 0, 5, 10,   1;
  6: 0, 6, 20,   6;
  7: 0, 7, 35,  21,  1;
  8: 0, 8, 56,  56,  8;
  9: 0, 9, 84, 126, 36, 1;
%F Formula
The row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1-x)*P(n-2,x) for n >= 2, with P(0,x) = 0, P(1,x) = x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = 2*v[i - 1] - (1 - x)*v[i - 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.

A034867: triangle, offset 0, from the row P1, low zeros dropped
%e Example
Triangle begins:
  0:  1;
  1:  2;
  2:  3,   1;
  3:  4,   4;
  4:  5,  10,   1;
  5:  6,  20,   6;
  6:  7,  35,  21,   1;
  7:  8,  56,  56,   8;
  8:  9,  84, 126,  36,  1;
  9: 10, 120, 252, 120, 10;
%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*P(n-1,x) - (1-x)*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = 2*x.
%t Mathematica
P[0, x_] := x;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*P[n - 2, x]];
Table[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = 2*v[i - 1] - (1 - x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A109447: reversed triangle, offset 1, from the row P1, low zeros dropped
%e Example
Triangle begins:
   1:  1;
   2:  2;
   3:  1,   3;
   4:  4,   4;
   5:  1,  10,   5;
   6:  6,  20,   6;
   7:  1,  21,  35,   7;
   8:  8,  56,  56,   8;
   9:  1,  36, 126,  84,  9;
  10: 10, 120, 252, 120, 10;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = 2*P(n-1,x) - (1-x)*P(n-2,x) for n >= 3, with P(1,x) = x, P(2,x) = 2*x.
%t Mathematica
P[1, x_] := x;
P[2, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*P[n - 2, x]];
Table[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = x; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = 2*v[i - 1] - (1 - x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = Vecrev(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-6fdf1329e5e61ed8
  • 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-6fdf1329e5e61ed8
  • 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-6fdf1329e5e61ed8

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 (11)

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. [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