Real-rooted atlas

A100861 → basis A013609 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2 - 2\,x + 3\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-9 - 12\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2426
310616
4464812106
516620060426
685613807201203076
738446972420084015856
8218204793638640134401680123516
911407627676825099210080015120757756
10703216196380021571201159200302400302406315976
11412549612548360152222409203040277200033264044203776
12273316249282451212961080094913280382536007983360665280391582456
131758494806435773769787720807915564803589185608648640086486403043809016
141241782816497435556884470305927863055200431855424013924310402421619201729728028496668656

Other recurrences

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

\(P_{n}(x) = \left(-3 - 2\,x + 3\,n + 2\,n x\right)\,P_{n-2}(x) + \left(1\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-72d93349d8bbe6d1 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-fc43d13584f4886f)
Fingerprint
f6-c16afd1c847e344e

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:      1;
  2:      4,      2;
  3:     10,      6;
  4:     46,     48,     12;
  5:    166,    200,     60;
  6:    856,   1380,    720,    120;
  7:   3844,   6972,   4200,    840;
  8:  21820,  47936,  38640,  13440,  1680;
  9: 114076, 276768, 250992, 100800, 15120;
%F Formula
The row polynomials satisfy P(n,x) = (3*n-2+(2*n-2)*x)*P(n-1,x) - (9+12*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[(3*n - 2 + (2*n - 2)*x)*P[n - 1, x] - (9 + 12*x + 4*x^2)*D[P[n - 1, x], x]];
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] = (3*n - 2 + (2*n - 2)*x)*v[i - 1] - (9 + 12*x + 4*x^2)*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 - 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