A161198 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	11	8														19
3	143	192	64													399
4	2145	4024	2496	512												9177
5	36465	85568	74624	28672	4096											229425
6	692835	1917512	2102400	1141760	307200	32768										6194475
7	14549535	45810432	59490496	40796160	15585280	3145728	262144									179639775
8	334639305	1170036216	1734764864	1414235648	684830720	197033984	31195136	2097152								5568833025
9	8365982625	31928019840	52729411328	49234010112	28434653184	10404495360	2356150272	301989888	16777216							183771489825
10	225881530875	928984396680	1679118264576	1751153563648	1161607716864	508398600192	146852020224	27002929152	2868903936	134217728						6432002143875
11	6550564395375	28747599750720	56126304846144	64216399462400	47695852298240	24036421140480	8325897388032	1957901107200	299221647360	26843545600	1073741824					237984079323375
12	203067496256625	943580107435320	1969896248236224	2439718822103552	1992302616944640	1126695873740800	450394188152832	127302113427456	24939079925760	3225923092480	248034361344	8589934592				9281379093611625
13	6701227376468625	32762683515418560	72555217051277952	96269891115307008	85263736936001536	53119384769003520	23876575198969856	7804123248328704	1841406544969728	305968101457920	33992518664192	2267742732288	68719476736			380536542838076625
14	234542958176401875	1200303742051398600	2801534064918076800	3949887925445968896	3754389921682509824	2541288362403135488	1260635210115973120	464156915283263488	126882215060570112	25440135910785024	3637482964910080	351311144943616	20547123544064	549755813888		16363071342037294875
15	8678089452526869375	46287582121312963200	113259190338380030400	168558125760845463552	170511530505820614656	124062788782376091648	66973809673516089344	27258887546408534016	8407897279507202048	1956342749183606784	338107956987953152	42098376082194432	3570732730679296	184717953466368	4398046511104	736338210391678269375

Technical details

Node id: n-ccc091ad88142f3d · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-f61108631909956f

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:            1;
   2:           11,            8;
   3:          143,          192,            64;
   4:         2145,         4024,          2496,           512;
   5:        36465,        85568,         74624,         28672,          4096;
   6:       692835,      1917512,       2102400,       1141760,        307200,        32768;
   7:     14549535,     45810432,      59490496,      40796160,      15585280,      3145728,       262144;
   8:    334639305,   1170036216,    1734764864,    1414235648,     684830720,    197033984,     31195136,     2097152;
   9:   8365982625,  31928019840,   52729411328,   49234010112,   28434653184,  10404495360,   2356150272,   301989888,   16777216;
  10: 225881530875, 928984396680, 1679118264576, 1751153563648, 1161607716864, 508398600192, 146852020224, 27002929152, 2868903936, 134217728;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n + 7 + 8*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; for(n = 2, N, my(i = n); v[i] = (2*n + 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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
  - ↑ Geometric basis map (elementary) (above)
  - ↑ Geometric basis map (elementary) (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.