Real-rooted atlas

A059473 → basis A013610 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 + 6\,x\right)\,P_{n-2}(x) + \left(10 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 10 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110616
210812636270
31160201611522164544
41246428776246609288129676484
51339203856324405682494806998477761287360
6143891249648327089408536047222628165054404665621668536
7154604806217036810652140810078041656847744191134083545856279936364719616
81661160967628186881524740544173235283212234252965498409601535448962435443216796166138873360
917848448009215009280210495110402791734336023687834400133329551044977542016118832832016460236810077696103328048384
10191773767681099584076802825600025604284777657604245639235202871958368961343005557124286464012889355571201098468864604661761739193001184
11206052526080129907748044837090366156806329773486080717351256512056681325573123185998645248127382161420835499020083265666267136725594112036279705629273680696320
1222139442749441522182148300847805061506048906751366963201156811445504001045673512001286866636126342433002928468864115215039912962849004969984473631556608475264143362176782336492727593157696
13237878629580801771308154798086068483017666561266474143582208179622900711936018281465151528961373564504340864771334017054336323715462018048100274344989696222823905177603363491506176309103091712130606940168293473020271616
142555901837803522049093570060288760503933149184017318062709028864270306349092679683058946487114854425881259131634176166307487919200008155348325250048303695728571289684535305738854417055125882265623574552698880199828618444878364164096139593754628553600
Technical details
Node id
n-9f60fedd71659dd7 · revision 0 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-a1fda25c3787eb76

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:         10,          6;
  2:        108,        126,          36;
  3:       1160,       2016,        1152,         216;
  4:      12464,      28776,       24660,        9288,        1296;
  5:     133920,     385632,      440568,      249480,       69984,        7776;
  6:    1438912,    4964832,     7089408,     5360472,     2262816,      505440,      46656;
  7:   15460480,   62170368,   106521408,   100780416,    56847744,    19113408,    3545856,     279936;
  8:  166116096,  762818688,  1524740544,  1732352832,  1223425296,   549840960,  153544896,   24354432,   1679616;
  9: 1784844800, 9215009280, 21049511040, 27917343360, 23687834400, 13332955104, 4977542016, 1188328320, 164602368, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (8+6*x)*P(n-2,x) + (10+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 10 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 10 + 6*x;
P[n_, x_] := P[n, x] = Expand[(8 + 6*x)*P[n - 2, x] + (10 + 6*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] = 10 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 6*x)*v[i - 2] + (10 + 6*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

Candidate equalities ?

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