A110552 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 3 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	4														7
2	10	26	16													52
3	36	136	164	64												400
4	136	664	1176	904	256											3136
5	528	3136	7264	8256	4624	1024										24832
6	2080	14496	41280	61760	51360	22560	4096									197632
7	8256	65920	222144	410880	451520	295296	106560	16384								1576960
8	32896	295808	1149568	2527616	3445120	2984576	1606528	491648	65536							12599296
9	131328	1312768	5774336	14694400	23873024	25704448	18357248	8390656	2228480	262144						100728832
10	524800	5771776	28329984	81831936	154205184	198245376	176203776	106973184	42471936	9961984	1048576					805568512
11	2098176	25176064	136360960	440524800	943933440	1409544192	1497581568	1132584960	597734400	209725440	44041216	4194304				6443499520
12	8390656	109074432	646035456	2307205120	5537157120	9412964352	11627556864	10519990272	6920939520	3229726720	1015044096	192940032	16777216			51543801856
13	33558528	469811200	3020169216	11812061184	31375421440	59797241856	84158300160	88586944512	69761691648	40601763840	16978812928	4831887360	838864896	67108864		412333637632
14	134225920	2013372416	13959282688	59326578688	172744073216	364680110080	575808110592	690966921216	633384001536	441447964672	230319964160	87242162176	22682902528	3623886848	268435456	3298601992192

Technical details

Node id: n-37f06382db0098d8 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-27961752b234205d

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:      3,       4;
  2:     10,      26,      16;
  3:     36,     136,     164,       64;
  4:    136,     664,    1176,      904,      256;
  5:    528,    3136,    7264,     8256,     4624,     1024;
  6:   2080,   14496,   41280,    61760,    51360,    22560,     4096;
  7:   8256,   65920,  222144,   410880,   451520,   295296,   106560,   16384;
  8:  32896,  295808, 1149568,  2527616,  3445120,  2984576,  1606528,  491648,   65536;
  9: 131328, 1312768, 5774336, 14694400, 23873024, 25704448, 18357248, 8390656, 2228480, 262144;

%F Formula

The row polynomials satisfy P(n,x) = -(8+16*x+8*x^2)*P(n-2,x) + (6+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(8 + 16*x + 8*x^2)*P[n - 2, x] + (6 + 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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 16*x + 8*x^2)*v[i - 2] + (6 + 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

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 A triangular array related to A077028 and distributing the values of A007582.?
  - 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