Real-rooted atlas

A019538 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2 - 5\,x - 3\,x^{2} + 2\,x^{3}\right)\,P_{n-2}(x) + \left(-x - 2\,x^{2} + x^{4}\right)\,P_{n-2}'(x) + \left(3 + 5\,x - x^{2}\right)\,P_{n-1}(x) + \left(x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2325
3718631
4158610824233
5313308707201202071
663114252208760540072021305
71273738270067560092400453605040249271
82551184612826854482410836001038240423360403203270713
9511368105775903522960103597201587600012489120435456036288047580151
10102311298225115402123052087280200194246640241315200160876800489888003628800760192505
1120473440581066500612192840067576080020641975203675505680383201280022153824005987520003991680013234467511
124095104240644547228676788024493067520019925484480480458563207107718464063767088000325321920007903526400479001600249383390393
138191314769018391191036648612003445774332017954576640056495222784011202410419201414779125760111307996800050802111360011208637440062270208005057242311031
141638394840227529394601948541868023317994700015373916157606150912768000157506725376002644771409280029103019545600203779255680008412705100800169997667840087178291200109820924003705
1532767285339783063879006102184664400153978781320012663237066480632010978799202036111261184004379116053312006365104986240006200483313024003909946360320001473313121280002746116172800013076743680002542685745501751
Technical details
Node id
n-c7dc4380efb2cd7f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0727a2b4e543a64a

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,      2;
   3:    7,     18,       6;
   4:   15,     86,     108,       24;
   5:   31,    330,     870,      720,      120;
   6:   63,   1142,    5220,     8760,     5400,       720;
   7:  127,   3738,   27006,    75600,    92400,     45360,      5040;
   8:  255,  11846,  128268,   544824,  1083600,   1038240,    423360,     40320;
   9:  511,  36810,  577590,  3522960, 10359720,  15876000,  12489120,   4354560,   362880;
  10: 1023, 112982, 2511540, 21230520, 87280200, 194246640, 241315200, 160876800, 48988800, 3628800;
%F Formula
The row polynomials satisfy P(n,x) = -(2+5*x+3*x^2-2*x^3)*P(n-2,x) - (x+2*x^2-x^4)*d/dx P(n-2,x) + (3+5*x-x^2)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(2 + 5*x + 3*x^2 - 2*x^3)*P[n - 2, x] - (x + 2*x^2 - x^4)*D[P[n - 2, x], x] + (3 + 5*x - x^2)*P[n - 1, x] + (x + x^2)*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 + 2*x; for(n = 3, N, my(i = n); v[i] = -(2 + 5*x + 3*x^2 - 2*x^3)*v[i - 2] - (x + 2*x^2 - x^4)*deriv(v[i - 2], x) + (3 + 5*x - x^2)*v[i - 1] + (x + x^2)*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

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.