Real-rooted atlas

A153656 → basis A013610

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-966 - 2898\,x + 483\,n + 1449\,n x\right)\,P_{n-2}(x) + \left(-4 - 6\,x + 2\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 46 + 69\,x\)
\(P_{2}(x) = 1058 + 3174\,x + 18\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
14669115
210583174184250
32433410950310953954243430
4559682335809251554073548071629428150
512872686965451452319622741749239281142991486517447510
629607177826646460028109496872868729120210412513553664683145820802423350
768096508947150133438727614462519746230466527028216121264147875121281170117943741103720706070

All 15 rows

Technical details
Node id
n-2f014dad76c8185f · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-b0048c17ed6c5e8a

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:             2;
  1:            46,             69;
  2:          1058,           3174,              18;
  3:         24334,         109503,          109539,              54;
  4:        559682,        3358092,         5155407,          354807,             162;
  5:      12872686,       96545145,       231962274,       174923928,         1142991,             486;
  6:     296071778,     2664646002,      8109496872,      8687291202,      1041251355,         3664683,           1458;
  7:    6809650894,    71501334387,    276144625197,    462304665270,    282161212641,      4787512128,       11701179,        4374;
  8:  156621970562,  1879463646744,   8544752299629,  17699665824225,  15042045777255,   2366601917463,    19696768623,    37224927,     13122;
  9: 3602305322926, 48631121859501, 259711183036404, 684460303783200, 889346414709018, 461023306641324, 14081956032564, 76121876736, 118038951, 39366;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (483*n-966+(1449*n-2898)*x)*P(n-2,x) + (2*n-4+(3*n-6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 46 + 69*x, P(2,x) = 1058 + 3174*x + 18*x^2.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 46 + 69*x;
P[2, x_] := 1058 + 3174*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((483*n - 966 + (1449*n - 2898)*x)*P[n - 2, x] + (2*n - 4 + (3*n - 6)*x)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 46 + 69*x; v[3] = 1058 + 3174*x + 18*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((483*n - 966 + (1449*n - 2898)*x)*v[i - 2] + (2*n - 4 + (3*n - 6)*x)*v[i - 1])/(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