Real-rooted atlas

A099093 → basis A013609 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9 + 18\,x\right)\,P_{n-2}(x) + \left(9 + 18\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 18\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
191827
290342324756
389150229396583221141
4882965934184032227448104976591219
587480813564301806055812245143824188956816533720
686678196490444464979210991570415179529611148451234012224462373353
78588349111340170617337612189749368834913967843845270880234684345661222003212930490971

All 15 rows

Technical details
Node id
n-1131397985fc0853 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f2c232606f089033

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:         9,          18;
  2:        90,         342,          324;
  3:       891,        5022,         9396,         5832;
  4:      8829,       65934,       184032,       227448,        104976;
  5:     87480,      813564,      3018060,      5581224,       5143824,       1889568;
  6:    866781,     9649044,     44649792,    109915704,     151795296,     111484512,      34012224;
  7:   8588349,   111340170,    617337612,   1897493688,    3491396784,    3845270880,    2346843456,     612220032;
  8:  85096170,  1259095266,   8135692488,  29982457800,   68922097776,  101188255968,   92649298176,   48365382528,  11019960576;
  9: 843160671, 14020240266, 103445108748, 444474105192, 1225560577824, 2248744643712, 2745568757952, 2150728972416, 980776491264, 198359290368;
%F Formula
The row polynomials satisfy P(n,x) = (9+18*x)*P(n-2,x) + (9+18*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 18*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 18*x;
P[n_, x_] := P[n, x] = Expand[(9 + 18*x)*P[n - 2, x] + (9 + 18*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 9 + 18*x; for(n = 2, N, my(i = n + 1); v[i] = (9 + 18*x)*v[i - 2] + (9 + 18*x)*v[i - 1]); 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

Same fingerprint ?

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