Real-rooted atlas

A161198: Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).

A161198 nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0) · coefficients of one sign · interlacing (weak) · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A161198
\(P_{n}(x) = \left(-3 + 2\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
338415
41546368105
510535234412816945
69453378380018404003210395
7103953904848556278408080115264135135
81351355284147093244590321607203113631361282027025

Show all 15 stored rows

Generated b-files: A161198 ?

Computed in your browser ?

Technical details
Node id
n-7c9c430277778c10 · revision 1 · JSON
Definition source
oeis_catalog:A161198
Fingerprint
f6-1d8809da307b5d7b

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 -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): 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        1;
   2:        1,         2;
   3:        3,         8,         4;
   4:       15,        46,        36,         8;
   5:      105,       352,       344,       128,       16;
   6:      945,      3378,      3800,      1840,      400,       32;
   7:    10395,     39048,     48556,     27840,     8080,     1152,      64;
   8:   135135,    528414,    709324,    459032,   160720,    31136,    3136,    128;
   9:  2027025,   8196480,  11696688,   8304128,  3328864,   788480,  109312,   8192,   256;
  10: 34459425, 143394210, 215236656, 164563552, 73198944, 20061888, 3435264, 357888, 20736, 512;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-3+2*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n - 3 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2*n - 3 + 2*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, v[i])))};
T(10)

Generated with polytool c48629374597.

A161198: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:        1;
  1:        1,         2;
  2:        3,         8,         4;
  3:       15,        46,        36,         8;
  4:      105,       352,       344,       128,       16;
  5:      945,      3378,      3800,      1840,      400,       32;
  6:    10395,     39048,     48556,     27840,     8080,     1152,      64;
  7:   135135,    528414,    709324,    459032,   160720,    31136,    3136,    128;
  8:  2027025,   8196480,  11696688,   8304128,  3328864,   788480,  109312,   8192,   256;
  9: 34459425, 143394210, 215236656, 164563552, 73198944, 20061888, 3435264, 357888, 20736, 512;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1+2*x)*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[(2*n - 1 + 2*x)*P[n - 1, x]];
Table[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] = (2*n - 1 + 2*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #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 >= 1, 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-6c4d7be2cbbf8954
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-6c4d7be2cbbf8954, e-84f434121cdb02d1
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-6c4d7be2cbbf8954, e-84f434121cdb02d1
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-6c4d7be2cbbf8954
  • Real-rooted. Every row P_n, n >= 1, 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-84f434121cdb02d1
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, 0]: (-inf, 0] -> (-inf, 0] through R_s, s = 2, and U_g
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-59e68d32a939d40d

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 7 proof steps

Same fingerprint (2)

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

References

  1. [Bre89] Francesco Brenti. Unimodal, log-concave and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc., 81(413):viii+106, 1989. doi:10.1090/memo/0413
  2. [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
  3. [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