Real-rooted atlas

A030528 → basis A013609 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,n\right)\,P_{n}(x) = \left(20 + 8\,x - 4\,n\right)\,P_{n-1}(x) + \left(21 + 20\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
21516435
35792488205
4216468372128161200
5819222623801248320327025
631051015213644963237607686441125
71177244982728286468834000105601792128240750
8446311951563696843959042619201095042822440962561409375
91692098332201807812226680018089449520323301767270492165128250625
106415203512808858924012343104115380007326336319827294720018201620480102448300000
112432187146595423988321264623816692606405152899227142080101166082613504445440450562048282753125
129221121606644641817620563278456643963297603381632642087317129387212830514176698880010700809830440961655265625
133495992424927863481558381616206989762181710160210025756814870058887837137923073605128855808018213888253132821299281929690093750
1413254313510181913843611923812784032566411631209040124713423369964054464602423296027687959049623900162489456644645683259105284587521638456726796875

Other recurrences

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

\(P_{n}(x) = \left(3 + 2\,x\right)\,P_{n-2}(x) + \left(3 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)
Technical details
Node id
n-732f971f7fa816ab · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-504abd919cd0fe17)
Fingerprint
f6-814ca2a6caa19aed

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,      2;
  2:     15,     16,       4;
  3:     57,     92,      48,       8;
  4:    216,    468,     372,     128,      16;
  5:    819,   2226,    2380,    1248,     320,     32;
  6:   3105,  10152,   13644,    9632,    3760,    768,     64;
  7:  11772,  44982,   72828,   64688,   34000,  10560,   1792,   128;
  8:  44631, 195156,  369684,  395904,  261920, 109504,  28224,  4096,  256;
  9: 169209, 833220, 1807812, 2266800, 1808944, 952032, 330176, 72704, 9216, 512;
%F Formula
The row polynomials satisfy (2*n+2)*P(n,x) = -(4*n-20-8*x)*P(n-1,x) + (21+20*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[(-(4*n - 20 - 8*x)*P[n - 1, x] + (21 + 20*x + 4*x^2)*D[P[n - 1, x], x])/(2*n + 2)]];
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 - 20 - 8*x)*v[i - 1] + (21 + 20*x + 4*x^2)*deriv(v[i - 1], x))/(2*n + 2)); 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