A091042 → basis A013609 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 32\,x - 64\,x^{2}\right)\,P_{n-2}(x) + \left(8 + 16\,x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 10 + 24\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	10	24														34
3	76	320	320													716
4	568	3360	6272	3584												13784
5	4240	32256	87552	98304	36864											259216
6	31648	294272	1047552	1757184	1351680	360448										4842784
7	236224	2595840	11435008	25559040	30031872	17039360	3407872									90305216
8	1763200	22356480	117381120	328048640	520519680	459669504	201850880	31457280								1683046784
9	13160704	189120512	1152827392	3868196864	7743078400	9340715008	6488588288	2281701376	285212672							31362601216
10	98232832	1577687040	10950770688	42891608064	103743750144	159124553728	152530059264	86067118080	24863834112	2550136832						584397750784
11	733219840	13015597056	101343707136	453878218752	1287680163840	2400190464000	2945819082752	2314450501632	1086559617024	263066746880	22548578304					10889285897216
12	5472827392	106402095104	918423240704	4629561999360	15075137028096	33103045197824	49627894120448	50239507529728	33108158054400	13175349051392	2716566814720	197568495616				202903086454784
13	40849739776	863256576000	8181019443200	45839758131200	168513267302400	426171820933120	755670974464000	938826163814400	801756309094400	451186314444800	154489973637120	27487790694400	1717986918400			3780745485193216
14	304906608640	6958909587456	71831439802368	442896850354176	1814316649021440	5194478310653952	10641499138031616	15693699367305216	16518986996908032	12110100525219840	5903501267828736	1760799152406528	273366078455808	14843406974976		70447582999158784
15	2275853910016	55789564461056	623031400267776	4192077764100096	18936372224065536	60554036511768576	140798614799646720	240601808769122304	301638681008013312	273638774519365632	174621292807323648	74612859060879360	19585875503480832	2678410325262336	127543348822016	1312667443460489216

Technical details

Node id: n-38ea897b21ea55e6 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-68435dc840e803c3

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:        1;
   2:       10,         24;
   3:       76,        320,         320;
   4:      568,       3360,        6272,        3584;
   5:     4240,      32256,       87552,       98304,        36864;
   6:    31648,     294272,     1047552,     1757184,      1351680,       360448;
   7:   236224,    2595840,    11435008,    25559040,     30031872,     17039360,      3407872;
   8:  1763200,   22356480,   117381120,   328048640,    520519680,    459669504,    201850880,    31457280;
   9: 13160704,  189120512,  1152827392,  3868196864,   7743078400,   9340715008,   6488588288,  2281701376,   285212672;
  10: 98232832, 1577687040, 10950770688, 42891608064, 103743750144, 159124553728, 152530059264, 86067118080, 24863834112, 2550136832;

%F Formula

The row polynomials satisfy P(n,x) = -(4+32*x+64*x^2)*P(n-2,x) + (8+16*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 10 + 24*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 10 + 24*x;
P[n_, x_] := P[n, x] = Expand[-(4 + 32*x + 64*x^2)*P[n - 2, x] + (8 + 16*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 10 + 24*x; for(n = 3, N, my(i = n); v[i] = -(4 + 32*x + 64*x^2)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    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 Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    ↑ 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