Real-rooted atlas

A269944 → basis A013611

nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
22121630
3106812864270
4100720155211522563780
517001264029264257928960102479380
6442003354408114247876483361286246440962381400
716354001258808031364448323886721558732836556804014081638497637400

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-afff5b8d53e95899 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-7a12baa115b5eeac

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 ?

No OEIS matches.

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:          1,           4;
  2:          2,          12,           16;
  3:         10,          68,          128,           64;
  4:        100,         720,         1552,         1152,         256;
  5:       1700,       12640,        29264,        25792,        8960,        1024;
  6:      44200,      335440,       811424,       787648,      336128,       62464,       4096;
  7:    1635400,    12588080,     31364448,     32388672,    15587328,     3655680,     401408,     16384;
  8:   81770000,   635945600,   1618574720,   1744891392,   908921088,   245133312,   34693120,   2424832,    65536;
  9: 5315050000, 41663544000, 107751139200, 119892239360, 66059436288, 19569349632, 3235586048, 296386560, 13959168, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (n^2-2*n+2+4*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[(n^2 - 2*n + 2 + 4*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] = (n^2 - 2*n + 2 + 4*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.

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-424ddc300423e142
  • 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-424ddc300423e142
  • 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-424ddc300423e142

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

No candidate equalities or related sequences.