Real-rooted atlas

A144438 → γ

nonnegative coefficients

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

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2134
311112
41292555
5167202270
6114510374291612
713034338606610708
8162116189513291256180701
91125956322339166274187670935
1012537187213193607334572605609356144019
111509560349010047170331140501746883761238643
1211021319047174882475326797802330695458235498609661170898
1312045159235062263779741935834321401866572614937774287680599407
14140929182306531014039721128972879944373831558934875390934302305346595566359286

Computed in your browser ?

Technical details
Node id
n-89920465e4889287 · revision 1 · JSON
Definition source
derived:gamma_polynomial
Fingerprint
f6-f4b7191086b91146

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): no

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;
  2: 1,    3;
  3: 1,   11;
  4: 1,   29,    25;
  5: 1,   67,   202;
  6: 1,  145,  1037,    429;
  7: 1,  303,  4338,   6066;
  8: 1,  621, 16189,  51329,  12561;
  9: 1, 1259, 56322, 339166, 274187;
%F Formula
The row polynomials satisfy P(n,x) = (1+(2*n-2)*x)*P(n-1,x) + (x-4*x^2)*d/dx P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n - 2)*x)*P[n - 1, x] + (x - 4*x^2)*D[P[n - 1, x], x] + x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (1 + (2*n - 2)*x)*v[i - 1] + (x - 4*x^2)*deriv(v[i - 1], x) + x*v[i - 2]); 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-da103189d8330c8e
  • 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-da103189d8330c8e
  • 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-da103189d8330c8e

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.

References

  1. [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