Real-rooted atlas

A132774: A natural number operator.

A023531 → 2D+1+2n

A132774 nonnegative coefficients only rational roots

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence from OEIS A132774
\(P_{n}(x) = \left(2\,x\right)\,P_{n-1}(x) + \left(-x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
20459
3006713
40008917
50000101121
600000121325
7000000141529

Show all 15 stored rows

Generated b-files: A004766 (row sums), A016813 (row sums), A132774 ?

Computed in your browser ?

Other recurrences

X ↦ x/(x+1) substitution · proved equal for all n (proof step)

\(\left(1 + x\right)\,P_{n}(x) = \left(4 + 11\,x + 9\,x^{2} + 2\,x^{3} - 2\,n - 6\,n x - 6\,n x^{2} - 2\,n x^{3}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 6\,x^{3} + 2\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-f4c1dee76f5a68b2 · revision 2 · JSON
Definition source
oeis_catalog:A132774
derived:mobius_x_over_x_plus_one (equal by e-2429e27d4b998677)
Fingerprint
f6-ccfaa2b0b0cc2c4e

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 ?

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: 2, 3;
  2: 0, 4, 5;
  3: 0, 0, 6, 7;
  4: 0, 0, 0, 8,  9;
  5: 0, 0, 0, 0, 10, 11;
  6: 0, 0, 0, 0,  0, 12, 13;
  7: 0, 0, 0, 0,  0,  0, 14, 15;
  8: 0, 0, 0, 0,  0,  0,  0, 16, 17;
  9: 0, 0, 0, 0,  0,  0,  0,  0, 18, 19;
%F Formula
The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] - x^2*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] = 1; v[2] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = 2*x*v[i - 1] - x^2*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.

A132774: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 2, 3;
   3: 0, 4, 5;
   4: 0, 0, 6, 7;
   5: 0, 0, 0, 8,  9;
   6: 0, 0, 0, 0, 10, 11;
   7: 0, 0, 0, 0,  0, 12, 13;
   8: 0, 0, 0, 0,  0,  0, 14, 15;
   9: 0, 0, 0, 0,  0,  0,  0, 16, 17;
  10: 0, 0, 0, 0,  0,  0,  0,  0, 18, 19;
%F Formula
The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) - x^2*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] - x^2*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 3*x; for(n = 3, N, my(i = n); v[i] = 2*x*v[i - 1] - x^2*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); 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-3fa49e62bb44b5e9
  • 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-3fa49e62bb44b5e9, e-cbd96c681ac3cbdc
  • 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-3fa49e62bb44b5e9, e-cbd96c681ac3cbdc
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-cbd96c681ac3cbdc

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Used in 9 proof steps

Same fingerprint (1)

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