Real-rooted atlas

A007318 → D+1 → x ↦ x+1

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(4 + x - n\right)\,P_{n-1}(x) + \left(2 + x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
286115
320249154
4488048121189
511224020080151648
62566727204001201812187
75761792235216807001682117290
812804608716862723360112022424124057
92816115202073621504141126048168028827178732
1061442816057600691205376028224100802400360301255879
11133126758415488021120019008011827251744158403300440331826686
1228672159744405504619520633600456192236544887042376044005283612657205
136144037273610383361757184201344016473609884164392961441443432057206243918503056
1413107286016026091524845568615014456376323843840197683276876822422448048728072842127103491

Generated b-files: A006234 (row sums), A103407 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-4b03a2849503b9e9 · revision 1 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-49d880ced4b36b3a

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:    3,     1;
  2:    8,     6,     1;
  3:   20,    24,     9,     1;
  4:   48,    80,    48,    12,     1;
  5:  112,   240,   200,    80,    15,    1;
  6:  256,   672,   720,   400,   120,   18,    1;
  7:  576,  1792,  2352,  1680,   700,  168,   21,   1;
  8: 1280,  4608,  7168,  6272,  3360, 1120,  224,  24,  1;
  9: 2816, 11520, 20736, 21504, 14112, 6048, 1680, 288, 27, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n-4-x)*P(n-1,x) + (2+x)*d/dx 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[-(n - 4 - x)*P[n - 1, x] + (2 + 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; for(n = 1, N, my(i = n + 1); v[i] = -(n - 4 - x)*v[i - 1] + (2 + 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.

A103407: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  3;
  2: 1,  6,   8;
  3: 1,  9,  24,   20;
  4: 1, 12,  48,   80,   48;
  5: 1, 15,  80,  200,  240,   112;
  6: 1, 18, 120,  400,  720,   672,   256;
  7: 1, 21, 168,  700, 1680,  2352,  1792,   576;
  8: 1, 24, 224, 1120, 3360,  6272,  7168,  4608,  1280;
  9: 1, 27, 288, 1680, 6048, 14112, 21504, 20736, 11520, 2816;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = -(n-4-x)*P(n-1,x) + (2+x)*d/dx 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[-(n - 4 - x)*P[n - 1, x] + (2 + x)*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = -(n - 4 - x)*v[i - 1] + (2 + x)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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-09eb0f66de37936d
  • 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-09eb0f66de37936d
  • 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-09eb0f66de37936d

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

Candidate equalities (1)

Definitions that agree on the stored rows without an all-n certificate.

  • A038207 → D+1 (15 rows compared; no all-index equality certificate was found)

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