Real-rooted atlas

A113953 → basis A013609 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(14 + 4\,x - 8\,n\right)\,P_{n-1}(x) + \left(33 + 28\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21516435
363102528225
4279588448144161475
51215318632401584368329625
6531916632211321388849448966462875
7232478451012884410639251152142402112128410625
81016554210927488727447204514561701763878448642562681875
9444447206663441999044885632357628817034885263361013761100851217515625
101943271100219682291058030520512261877441508992058993921541120256768245761024114396875
1184964954813003812224163618349610418060451212217132857849984191357444324096634368542722048747140625
1237149111229287996640536336106973697611880142569230135045136486402059514885894886411725824153600011878440964879671875
131624263031084928418330615496860802267687521128208660051504042227581441991366144691237120174177792309227523657728258048819231869765625
1471017357551039622801684869174033833557440461308716004513393958432643250368177099182087233942784221115801649724313679675392858931255705616384208145546875

Other recurrences

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

\(P_{n}(x) = \left(6 + 4\,x\right)\,P_{n-2}(x) + \left(3 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)
Technical details
Node id
n-a9609862a68b4355 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-f068516af8b6b455)
Fingerprint
f6-46d787fbc997ccf4

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:      3,       2;
  2:     15,      16,       4;
  3:     63,     102,      52,       8;
  4:    279,     588,     448,     144,      16;
  5:   1215,    3186,    3240,    1584,     368,      32;
  6:   5319,   16632,   21132,   13888,    4944,     896,     64;
  7:  23247,   84510,  128844,  106392,   51152,   14240,   2112,    128;
  8: 101655,  421092,  748872,  744720,  451456,  170176,  38784,   4864,   256;
  9: 444447, 2066634, 4199904, 4885632, 3576288, 1703488, 526336, 101376, 11008, 512;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = -(8*n-14-4*x)*P(n-1,x) + (33+28*x+4*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[(-(8*n - 14 - 4*x)*P[n - 1, x] + (33 + 28*x + 4*x^2)*D[P[n - 1, x], x])/(2*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] = (-(8*n - 14 - 4*x)*v[i - 1] + (33 + 28*x + 4*x^2)*deriv(v[i - 1], x))/(2*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