Real-rooted atlas

A207824 → x ↦ x+1 → basis A008277

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(7 + x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1718
24815164
3329167241521
4225516503823414322
51545615288506472145136575
610593713611060482122401215571316042
772610311795296753841821612556018987012790706
84976784100262257197503248475446210648279280784125198543
93411138584007055741283683186288275423601039556847003982991232780388
10233802911696141600743964864390272434114366736199687532137849140364546611512200801093
11160250899257189286567317697008461482632216407641703529522164767609040989632222557305132121299682110
1210983760033466577966407083423686453075687794225488654565856222009970603537104921208742732233902095481501211039869582
137528381123937852720449767684823140859697628848229947244364092470577348184263919002435401486216108783128443375012041224716912140669816741
14516002918640305684365057563993270503056593535428434386862430306614032631498073310430685115241709149656696059094212791552135525772150015457189122227169638302
Technical details
Node id
n-34199e497f5ec9ac · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-a30094de5c079a98

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        7,        1;
  2:       48,       15,        1;
  3:      329,      167,       24,        1;
  4:     2255,     1650,      382,       34,       1;
  5:    15456,    15288,     5064,      721,      45,       1;
  6:   105937,   136110,    60482,    12240,    1215,      57,     1;
  7:   726103,  1179529,   675384,   182161,   25560,    1898,    70,    1;
  8:  4976784, 10026225,  7197503,  2484754,  462106,   48279,  2807,   84,  1;
  9: 34111385, 84007055, 74128368, 31862882, 7542360, 1039556, 84700, 3982, 99, 1;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (7+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 7 + x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (7 + x)*P[n - 1, x] + x*D[P[n - 1, x], 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] = 7 + x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (7 + x)*v[i - 1] + x*deriv(v[i - 1], x)); 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)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023