Real-rooted atlas

A063967 → basis A013610 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(9 + 3\,x - 2\,n\right)\,P_{n-1}(x) + \left(15 + 14\,x + 3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2538
33033972
417528515327640
5102522051764621815696
6600016050170558991234924350688
735125112350148860104517409868505729451072
8205625765375121477510652855571991737452988921874014080
9120375051108759451350993991565003312705805699840102789656135721216
107046875336093757095937587024375683032503557034012287295271406734773319683317882368
114125312521835312551821437572605362566491077541579146817976411053045685102220381161297590492828828672
1224150000014047125003701756250583291125061053041254456451115231434244985495297522012811137614213383818517714725173688320
1314137656258963587500259690500004545473625053527969125446718576152708726054412022312977387574672588492799713581270012577437531441224020135936
14827632812556807296875179458931250345450656250452029308750424541363400294392941155152591138739591111720361690033555834658810554827255754092095715943231993550594048
154845046875035792470312512244525593752570810906250370050256687538627379387003015040671540178750786367780877494895327788707405871364259440132815963251693171026132328809478296917740565839872

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(5 + 3\,x\right)\,P_{n-2}(x) + \left(5 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 3\,x\)
Technical details
Node id
n-c551333e3802137a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-24f0dfd02baa8b41)
Fingerprint
f6-69e7feb8617913a4

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:       5,        3;
   3:      30,       33,        9;
   4:     175,      285,      153,       27;
   5:    1025,     2205,     1764,      621,       81;
   6:    6000,    16050,    17055,     8991,     2349,      243;
   7:   35125,   112350,   148860,   104517,    40986,     8505,      729;
   8:  205625,   765375,  1214775,  1065285,   557199,   173745,    29889,    2187;
   9: 1203750,  5110875,  9451350,  9939915,  6500331,  2705805,   699840,  102789,   6561;
  10: 7046875, 33609375, 70959375, 87024375, 68303250, 35570340, 12287295, 2714067, 347733, 19683;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(2*n-9-3*x)*P(n-1,x) + (15+14*x+3*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 9 - 3*x)*P[n - 1, x] + (15 + 14*x + 3*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n - 9 - 3*x)*v[i - 1] + (15 + 14*x + 3*x^2)*deriv(v[i - 1], x))/(n - 1)); 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. [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
  2. [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