Real-rooted atlas

A059473 → basis A013609 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + 16\,x\right)\,P_{n-2}(x) + \left(8 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
181624
270272256598
360835206656409614880
452844064011545614336065536370276
54592044051216701443125248288358410485769213984
639906445872002175232054431744757596165557452816777216229281688
73468032464604162644828168289648641544290304170917888010401873922684354565705468160

All 15 rows

Technical details
Node id
n-4260335b964afa5a · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-e56348271b1ee2e5

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:         8,         16;
  2:        70,        272,         256;
  3:       608,       3520,        6656,         4096;
  4:      5284,      40640,      115456,       143360,        65536;
  5:     45920,     440512,     1670144,      3125248,      2883584,      1048576;
  6:    399064,    4587200,    21752320,     54431744,     75759616,     55574528,     16777216;
  7:   3468032,   46460416,   264482816,    828964864,   1544290304,   1709178880,   1040187392,    268435456;
  8:  30138640,  461080064,  3063138304,  11538071552,  26943225856,  39927676928,  36658216960,  19058917376,   4294967296;
  9: 261917312, 4505109760, 34212651008, 150520299520, 422684131328, 785476747264, 965696552960, 757256421376, 343597383680, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (6+16*x)*P(n-2,x) + (8+16*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 16*x;
P[n_, x_] := P[n, x] = Expand[(6 + 16*x)*P[n - 2, x] + (8 + 16*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] = 8 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (6 + 16*x)*v[i - 2] + (8 + 16*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

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