Real-rooted atlas

A269947 → basis A007318

nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
22316
3182912160
45048303654011740
532760544542455529651051114840
641277606893964314838439814516195231114584680
789572392015001179486900932928954584939124606632244813179460240

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-b096a5459ca672a4 · revision 1 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-759e2feadaf031a1

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,               1;
  2:               2,               3,               1;
  3:              18,              29,              12,              1;
  4:             504,             830,             365,             40,            1;
  5:           32760,           54454,           24555,           2965,          105,           1;
  6:         4127760,         6893964,         3148384,         398145,        16195,         231,         1;
  7:       895723920,      1500117948,       690093292,       89545849,      3912460,       66322,       448,      1;
  8:    308129028480,    516936298032,    238892210396,    31493865348,   1435432089,    26727228,    220434,    792,    1;
  9: 158070191610240, 265496449918896, 123068640231180, 16395245133920, 767870527005, 15146500053, 139809870, 626730, 1305, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n^3-3*n^2+3*n+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^3 - 3*n^2 + 3*n + 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^3 - 3*n^2 + 3*n + 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-0e122e2df114c72b
  • 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-0e122e2df114c72b
  • 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-0e122e2df114c72b

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.