A172431 → basis A013609 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 24\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(7 + 8\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 7 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	7	8														15
2	40	88	48													176
3	217	696	736	256												1905
4	1159	4856	7536	5120	1280											19951
5	6160	31792	64800	65024	32000	6144										205920
6	32689	200304	505024	668928	489216	186368	28672									2111201
7	173383	1229672	3692832	6073600	5890560	3354624	1032192	131072								21577935
8	919480	7407496	25811664	50712064	61285120	46485504	21504000	5505024	589824							220220176
9	4875913	43980072	174419872	398512640	576825600	546936832	338362368	130940928	28508160	2621440						2245983825
10	25856071	258132824	1147983120	2990539008	5044238080	5742557184	4454281216	2314076160	765591552	144179200	11534336					22898968751
11	137109280	1500935776	7397629248	21641266176	41687526912	55409356800	51719471104	33782628352	15058796544	4331143168	715128832	50331648				233431323840
12	727060321	8659683552	46850157952	152073326080	329404008960	500772360192	546693439488	430623817728	241975951360	94091345920	23841472512	3489660928	218103808			2379420388801
13	3855438727	49635222728	292433703744	1042985076224	2509474472960	4295193477120	5370732724224	4956053569536	3364998414336	1653533573120	568296734720	128245039104	16793993216	939524096		24253170963855
14	20444528200	282903469240	1803000916080	7008746501120	18546203486720	35296329449472	49747435143168	52649622110208	41943567237120	24950496296960	10861964951552	3335025328128	676339908608	79859548160	4026531840	247205965406576

Technical details

Node id: n-37be3118d4ffb268 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-e7d980e9e7184757

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:       7,        8;
  2:      40,       88,        48;
  3:     217,      696,       736,       256;
  4:    1159,     4856,      7536,      5120,      1280;
  5:    6160,    31792,     64800,     65024,     32000,      6144;
  6:   32689,   200304,    505024,    668928,    489216,    186368,     28672;
  7:  173383,  1229672,   3692832,   6073600,   5890560,   3354624,   1032192,    131072;
  8:  919480,  7407496,  25811664,  50712064,  61285120,  46485504,  21504000,   5505024,   589824;
  9: 4875913, 43980072, 174419872, 398512640, 576825600, 546936832, 338362368, 130940928, 28508160, 2621440;

%F Formula

The row polynomials satisfy P(n,x) = -(9+24*x+16*x^2)*P(n-2,x) + (7+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 7 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(9 + 24*x + 16*x^2)*P[n - 2, x] + (7 + 8*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] = 7 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(9 + 24*x + 16*x^2)*v[i - 2] + (7 + 8*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 basis_map_A013609 of Even row Pascal-square read by antidiagonals.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Even row Pascal-square read by antidiagonals.?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Even row Pascal-square read by antidiagonals.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Even row Pascal-square read by antidiagonals.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Even row Pascal-square read by antidiagonals.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Even row Pascal-square read by antidiagonals.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023