Real-rooted atlas

A081577 → basis A013609 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(11 + 4\,x - 10\,n - 2\,n x + 3\,n^{2}\right)\,P_{n-1}(x) + \left(56 + 49\,x + 8\,x^{2} - 20\,n - 16\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(30 + 44\,x + 19\,x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
22220446
3112152648336
45801040664176162476
529926680572823364483218216
6154484126444416245127264108864134056
77974424800032128022380890112208642560128986496

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(6 + 4\,x\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)
Technical details
Node id
n-76f18d55a500291a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-348e124ea2bb9b77)
Fingerprint
f6-f853454275d86d84

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:      22,      20,        4;
  3:     112,     152,       64,        8;
  4:     580,    1040,      664,      176,      16;
  5:    2992,    6680,     5728,     2336,     448,      32;
  6:   15448,   41264,    44416,    24512,    7264,    1088,     64;
  7:   79744,  248000,   321280,   223808,   90112,   20864,   2560,    128;
  8:  411664, 1460864,  2212672,  1862528,  949696,  299264,  56704,   5888,   256;
  9: 2125120, 8473760, 14692096, 14503424, 8959744, 3582080, 924160, 147968, 13312, 512;
%F Formula
The row polynomials satisfy n*P(n,x) = (3*n^2-10*n+11-(2*n-4)*x)*P(n-1,x) - (20*n-56+(16*n-49)*x+(2*n-8)*x^2)*d/dx P(n-1,x) + (30+44*x+19*x^2+2*x^3)*d^2/dx^2 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[((3*n^2 - 10*n + 11 - (2*n - 4)*x)*P[n - 1, x] - (20*n - 56 + (16*n - 49)*x + (2*n - 8)*x^2)*D[P[n - 1, x], x] + (30 + 44*x + 19*x^2 + 2*x^3)*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] = ((3*n^2 - 10*n + 11 - (2*n - 4)*x)*v[i - 1] - (20*n - 56 + (16*n - 49)*x + (2*n - 8)*x^2)*deriv(v[i - 1], x) + (30 + 44*x + 19*x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), 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