Real-rooted atlas

A113953 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(8 + 3\,x - 4\,n\right)\,P_{n-1}(x) + \left(16 + 16\,x + 3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1437
22430963
312824014427539
470417281548594814655
5384011712139688100226824340131
6209927641611376088344375848262729346087
71146884853768663048432644821121613522916021872984443
862668830228486289920736128052889762383344656100100602656125736319
93424256185425924407091260258816521562242958379210975824256316434117219683221936435
10187105281123860483003678724698777604758808323256044481522035364792154497102801141614590491913863511
111022361606745620482002452480352711065640919592963279667968185093683273465704020071411235901792377913617714716504154667
12558628864401630822413110755328256770293763357548236830854670336204140062089788032224337203907281314409613014399612400290531441142323171823
1330524047362375103283284558938112182271983616265231355904274934227968208705538304117377024832488659080961486019260832056521124641251404038951615943231227320368099
14166786498561396435845125384727429121266845515776203040740966423437212364802008064908800129641300928063329921280023277588192063323202816123535231921632586752130734486478296910583766982215

Other recurrences

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

\(P_{n}(x) = \left(8 + 6\,x\right)\,P_{n-2}(x) + \left(4 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 3\,x\)
Technical details
Node id
n-b716643bd0e1e1b1 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-6e343c6e9ae63867)
Fingerprint
f6-3f387007411de7e8

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:       4,        3;
  2:      24,       30,        9;
  3:     128,      240,      144,       27;
  4:     704,     1728,     1548,      594,       81;
  5:    3840,    11712,    13968,     8100,     2268,      243;
  6:   20992,    76416,   113760,    88344,    37584,     8262,      729;
  7:  114688,   485376,   866304,   843264,   482112,   161352,    29160,    2187;
  8:  626688,  3022848,  6289920,  7361280,  5288976,  2383344,   656100,  100602,   6561;
  9: 3424256, 18542592, 44070912, 60258816, 52156224, 29583792, 10975824, 2563164, 341172, 19683;
%F Formula
The row polynomials satisfy n*P(n,x) = -(4*n-8-3*x)*P(n-1,x) + (16+16*x+3*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 - 3*x)*P[n - 1, x] + (16 + 16*x + 3*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(4*n - 8 - 3*x)*v[i - 1] + (16 + 16*x + 3*x^2)*deriv(v[i - 1], x))/(n)); 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

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