Real-rooted atlas

A137320 → reverse

nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
122
2426
3812424
416484412120
53216028020048720
6644801360180010962405040
71281344560011760129927056144040320

Show all 15 stored rows

Generated b-files: A125553 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-3df3926aefc1fd76 · revision 1 · JSON
Definition source
derived:reversal
Fingerprint
f6-21d4cbc8ebff768c

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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;
  2:   4,    2;
  3:   8,   12,     4;
  4:  16,   48,    44,     12;
  5:  32,  160,   280,    200,     48;
  6:  64,  480,  1360,   1800,   1096,     240;
  7: 128, 1344,  5600,  11760,  12992,    7056,   1440;
  8: 256, 3584, 20608,  62720, 108304,  105056,  52272,  10080;
  9: 512, 9216, 69888, 290304, 718368, 1076544, 944992, 438336, 80640;
%F Formula
The row polynomials satisfy P(n,x) = (2+(n-1)*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)*x)*P[n - 1, 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] = (2 + (n - 1)*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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A125553: reversed triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1:      2;
   2:      2,       4;
   3:      4,      12,       8;
   4:     12,      44,      48,       16;
   5:     48,     200,     280,      160,      32;
   6:    240,    1096,    1800,     1360,     480,      64;
   7:   1440,    7056,   12992,    11760,    5600,    1344,     128;
   8:  10080,   52272,  105056,   108304,   62720,   20608,    3584,    256;
   9:  80640,  438336,  944992,  1076544,  718368,  290304,   69888,   9216,   512;
  10: 725760, 4106304, 9381600, 11578880, 8618400, 4049472, 1209600, 222720, 23040, 1024;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+(n-1)*x)*P(n-1,x) for n >= 2, with P(1,x) = 2.
%t Mathematica
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(2 + (n - 1)*x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = (2 + (n - 1)*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); 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-a28ae8b138b0cf1e
  • 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-a28ae8b138b0cf1e
  • 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-a28ae8b138b0cf1e

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 (3)

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