Real-rooted atlas

A111577 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 + x + 2\,x^{2}\right)\,P_{n-2}(x) + \left(-3\,x + 6\,x^{2}\right)\,P_{n-2}'(x) + \left(5 - x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2213
367114
4224515183
586295171261579
6342198117954654014624
7136613527182917190103557141467
8546293325184275103621221902016771410967
9218466478151848211143778643024157302357010014446872
108738245128611850395519515945785862014394871299905886126152054253
113495263150264718512693126135043013821109533473979410858726724091801551654399771
121398102220169005185161883534696489412370311130739415622118505079103811075085301369518718781970234
135592406153978493518517586451457955661463995017478115759973944320992782036368838023817717909205197012221125167041709
14223696221077290214118518145691560202628622566505899460032739748341383123760720118499615529961843325053248215442572749526011886481504015
158947848675387945367185183693877178916523840701097192475515566737966705412082290142933366170307360384309439802487114630100488388251251037401301129954169719894
Technical details
Node id
n-dfa4a24822c608f2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-392da80327ba55d9

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:     2,       1;
   3:     6,       7,        1;
   4:    22,      45,       15,        1;
   5:    86,     295,      171,       26,       1;
   6:   342,    1981,     1795,      465,      40,       1;
   7:  1366,   13527,    18291,     7190,    1035,      57,      1;
   8:  5462,   93325,   184275,   103621,   22190,    2016,     77,    1;
   9: 21846,  647815,  1848211,  1437786,  430241,   57302,   3570,  100,   1;
  10: 87382, 4512861, 18503955, 19515945, 7858620, 1439487, 129990, 5886, 126, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(4-x-2*x^2)*P(n-2,x) - (3*x-6*x^2)*d/dx P(n-2,x) + (5-x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[-(4 - x - 2*x^2)*P[n - 2, x] - (3*x - 6*x^2)*D[P[n - 2, x], x] + (5 - x)*P[n - 1, x] + 3*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] = 2 + x; for(n = 3, N, my(i = n); v[i] = -(4 - x - 2*x^2)*v[i - 2] - (3*x - 6*x^2)*deriv(v[i - 2], x) + (5 - x)*v[i - 1] + 3*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.