Real-rooted atlas

A100862 → x ↦ x+1 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(2\right)\,P_{n}(x) = \left(10 + 4\,x - 5\,n - 2\,n x + n^{2}\right)\,P_{n-1}(x) + \left(26 + 18\,x + 2\,x^{2} - 8\,n - 4\,n x\right)\,P_{n-1}'(x) + \left(16 + 16\,x + 4\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
2117119
34542121100
4201246105181571
596314558252152513484
6489987936210217539033122501
7262535451646053202654935651421152716

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(-2 - x + 2\,n + n x\right)\,P_{n-2}(x) + \left(3 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)
Technical details
Node id
n-913437c43d2bcfce · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-55e05e8a65d57e46)
Fingerprint
f6-68e42cf18f57debe

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,       1;
  2:     11,       7,       1;
  3:     45,      42,      12,       1;
  4:    201,     246,     105,      18,      1;
  5:    963,    1455,     825,     215,     25,      1;
  6:   4899,    8793,    6210,    2175,    390,     33,     1;
  7:  26253,   54516,   46053,   20265,   4935,    651,    42,    1;
  8: 147345,  347196,  341166,  180768,  55755,  10080,  1022,   52,  1;
  9: 862083, 2271213, 2543670, 1576134, 589113, 135891, 19026, 1530, 63, 1;
%F Formula
The row polynomials satisfy 2*P(n,x) = (n^2-5*n+10-(2*n-4)*x)*P(n-1,x) - (8*n-26+(4*n-18)*x-2*x^2)*d/dx P(n-1,x) + (16+16*x+4*x^2)*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[((n^2 - 5*n + 10 - (2*n - 4)*x)*P[n - 1, x] - (8*n - 26 + (4*n - 18)*x - 2*x^2)*D[P[n - 1, x], x] + (16 + 16*x + 4*x^2)*D[P[n - 1, x], {x, 2}])/(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] = ((n^2 - 5*n + 10 - (2*n - 4)*x)*v[i - 1] - (8*n - 26 + (4*n - 18)*x - 2*x^2)*deriv(v[i - 1], x) + (16 + 16*x + 4*x^2)*deriv(deriv(v[i - 1], x), x))/(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