A308737 → basis A013610

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(-8 - 24\,x - 18\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 9\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 4 + 9\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	9														13
2	16	66	63													145
3	64	372	684	405												1525
4	256	1896	5076	5886	2511											15625
5	1024	9168	31968	54648	46008	15309										158125
6	4096	42912	183600	412560	515160	339714	92583									1590625
7	16384	196416	993600	2758320	4548960	4464396	2417364	557685								15953125
8	65536	884352	5156928	17013024	34791120	45219384	36517068	16765542	3352671							159765625
9	262144	3931392	25943040	99042048	241387776	389896416	417711168	286409520	114108912	20135709						1598828125
10	1048576	17299968	127381248	551950848	1560238848	3008892096	4011692832	3653374752	2175837552	765550602	120873303					15994140625
11	4194304	75494400	613370880	2972401920	9553904640	21400267392	34106002560	38691004320	30629897280	16120770660	5077977804	725416965				159970703125
12	16777216	327149568	2906551296	15570593280	56053451520	142934761728	264847169664	359432365248	354700257120	248288447880	117049525956	33373431918	4353033231			1599853515625
13	67108864	1409273856	13589323776	79723505664	317646489600	908090548992	1917071152128	3026941809408	3575561501952	3121511223600	1958032717344	835837650216	217665479016	26119793709		15999267578125
14	268435456	6039773184	62813417472	400434278400	1748950714368	5538330091008	13117068087552	23610674520576	32464617419520	33940224530784	26561873731536	15091952217264	5885853626952	1410513501330	156723545223	159996337890625

Technical details

Node id: n-6f2154a01224373b · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-527aea5dab844fde

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      1;
  1:      4,       9;
  2:     16,      66,       63;
  3:     64,     372,      684,      405;
  4:    256,    1896,     5076,     5886,      2511;
  5:   1024,    9168,    31968,    54648,     46008,     15309;
  6:   4096,   42912,   183600,   412560,    515160,    339714,     92583;
  7:  16384,  196416,   993600,  2758320,   4548960,   4464396,   2417364,    557685;
  8:  65536,  884352,  5156928, 17013024,  34791120,  45219384,  36517068,  16765542,   3352671;
  9: 262144, 3931392, 25943040, 99042048, 241387776, 389896416, 417711168, 286409520, 114108912, 20135709;

%F Formula

The row polynomials satisfy P(n,x) = -(8+24*x+18*x^2)*P(n-2,x) + (6+9*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 9*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 9*x;
P[n_, x_] := P[n, x] = Expand[-(8 + 24*x + 18*x^2)*P[n - 2, x] + (6 + 9*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] = 4 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 24*x + 18*x^2)*v[i - 2] + (6 + 9*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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is t…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[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