A085478 → basis A013611 → basis A013611

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0	1														1
1	1														1
2	6	16													22
3	41	208	256												505
4	281	2096	5120	4096											11593
5	1926	18960	69120	110592	65536										266134
6	13201	161440	782080	1875968	2228224	1048576									6109489
7	90481	1322336	7988480	25534464	45547520	42991616	16777216								140252113
8	620166	10542608	76294656	304680960	725155840	1028653056	805306368	268435456							3219689110
9	4250681	82398576	694755840	3327946752	9905438720	18760073216	22078816256	14763950080	4294967296						73912597417
10	29134601	634258320	6105373440	34107039744	121860063232	288778878976	453907578880	456340275200	266287972352	68719476736					1696770051481
11	199691526	4823563280	52190991360	333107306496	1388827639808	3952453091328	7775736299520	10442139238400	9161165242368	4741643894784	1099511627776				38951798586646
12	1368706081	36325749056	436408578560	3132699967488	14929650319360	49599634997248	117215495979008	197050415185920	230936096538624	179701431664640	83562883710976	17592186044416			894194597441377
13	9381251041	271355977408	3583881043456	28578329722880	153241924075520	582119396999168	1606326895509504	3244358702727168	4760198153502720	4948145922375680	3459063580983296	1460151441686528	281474976710656		20527523942565025
14	64300051206	2013266109456	28992454364160	254257704787968	1515017093775360	6477106929205248	20440983124574208	48214690832056320	85000190221615104	110620490481008640	103300216941182976	65548485201494016	25332747903959040	4503599627370496	471238856081554198

Technical details

Node id: n-3bbccf23445ea940 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-f01e0b1962cfad1f

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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:       1;
  2:       6,       16;
  3:      41,      208,       256;
  4:     281,     2096,      5120,       4096;
  5:    1926,    18960,     69120,     110592,      65536;
  6:   13201,   161440,    782080,    1875968,    2228224,     1048576;
  7:   90481,  1322336,   7988480,   25534464,   45547520,    42991616,    16777216;
  8:  620166, 10542608,  76294656,  304680960,  725155840,  1028653056,   805306368,   268435456;
  9: 4250681, 82398576, 694755840, 3327946752, 9905438720, 18760073216, 22078816256, 14763950080, 4294967296;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (7 + 16*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (7 + 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 - 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operato…?
    
    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
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] w…?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operato…?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (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