Real-rooted atlas

A094816 → x ↦ x+1 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
3118120
44957151122
5261424174241884
61631342519304103517432
7117433033621855632082548171128
89590129455325916195235168351491631763240
98768093123632325150014529763251503886424928019071504
1088776913600396943319844228312846195546935046807243924991118248128
1198641011448816600613218285372712320119033670217415522365650154560589512011676689664
12119355623360220337219213892435634830256523343292505010311466588166254176102773658525143125684730656
1315624736141865870794481466989115461130352005204705685869511614763952179264223617795131211454795472120119461681422600082880
14220048367319132875360265724699018665102105152890126979103564725273339763983484696709645642559780438083217226833757690541630219517430873838208
153317652307271216832275796644387990134973940998349172144210754850215916568246676992131636817612317593517884816007644653999030032425234811207024217492241139032215889536
Technical details
Node id
n-3382a8d6902cee06 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-777ea511f0ccda38

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       3,        1;
   3:      11,        8,        1;
   4:      49,       57,       15,        1;
   5:     261,      424,      174,       24,       1;
   6:    1631,     3425,     1930,      410,      35,      1;
   7:   11743,    30336,    21855,     6320,     825,     48,     1;
   8:   95901,   294553,   259161,    95235,   16835,   1491,    63,    1;
   9:  876809,  3123632,  3251500,  1452976,  325150,  38864,  2492,   80,  1;
  10: 8877691, 36003969, 43319844, 22831284, 6195546, 935046, 80724, 3924, 99, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n-2)*P(n-2,x) + (n+1+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[-(n - 2)*P[n - 2, x] + (n + 1 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = -(n - 2)*v[i - 2] + (n + 1 + 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, v[i])))};
T(10)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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