A257180 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(3 + 12\,x + 15\,x^{2} + 6\,x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 9 + 42\,x + 42\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	9	42	42															93
2	27	222	504	336														1089
3	81	1050	4074	6048	3024													14277
4	243	4686	27720	68880	75600	30240												207369
5	729	20202	171402	635040	1149120	997920	332640											3307053
6	2187	85182	998424	5181456	13759200	19474560	13970880	3991680										57463569
7	6561	353850	5588394	39076128	142947504	293388480	339958080	207567360	51891840									1080778197
8	19683	1454766	30418920	279163920	1352317680	3776582880	6286896000	6155170560	3269185920	726485760								21877696089
9	59049	5937162	162277962	1918334880	11978699040	43735839840	98124475680	136994457600	115978262400	54486432000	10897286400							474282062013
10	177147	24102942	852949944	12808232976	101115907200	469673709120	1364214519360	2547438284160	3056688835200	2277532857600	958961203200	174356582400						10963667361249
11	531441	97474650	4433470314	83672997408	823085447184	4768548744960	17444341474560	41844957073920	66740274881280	70124037984000	46629488505600	17784371404800	2964061900800					269211371890917
12	1594323	393087246	22849674120	537505746960	6514655106960	46379244431520	209453109465600	627781668433920	1278127465655040	1775852304145920	1658043920332800	995053015756800	346795242393600	53353114214400				6997914490039209
13	4782969	1581914922	116999981322	3407831874720	50440137471360	436180506521760	2395249673937120	8791253997926400	22220563811846400	39204012168921600	48192355526515200	40459444945920000	22117829903769600	7095964190515200	1013709170073600			191980530451972173
14	14348907	6356357502	596073311064	21382991098896	383751449172000	3993845426887680	26355232637173440	116655536066509440	358712503903353600	781536039392908800	1214560804378521600	1337509917667123200	1019578012637184000	511656365316096000	152056375511040000	20274183401472000		5543294553222558129
15	43046721	25511523450	3024861057834	133066533081888	2878707064094064	35788277906821440	281129393430325440	1482818152311175680	5462359511801512320	14400407527361856000	27512331047814412800	38158111717396070400	38031338789898854400	26538906072526848000	12306429324693504000	3406062811447296000	425757851430912000	168044455302032392437

Technical details

Node id: n-1bebae31ffd0b112 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-09455fa73ad50cf5

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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:      9,       42,        42;
   2:     27,      222,       504,         336;
   3:     81,     1050,      4074,        6048,         3024;
   4:    243,     4686,     27720,       68880,        75600,        30240;
   5:    729,    20202,    171402,      635040,      1149120,       997920,        332640;
   6:   2187,    85182,    998424,     5181456,     13759200,     19474560,      13970880,       3991680;
   7:   6561,   353850,   5588394,    39076128,    142947504,    293388480,     339958080,     207567360,      51891840;
   8:  19683,  1454766,  30418920,   279163920,   1352317680,   3776582880,    6286896000,    6155170560,    3269185920,     726485760;
   9:  59049,  5937162, 162277962,  1918334880,  11978699040,  43735839840,   98124475680,  136994457600,  115978262400,   54486432000,  10897286400;
  10: 177147, 24102942, 852949944, 12808232976, 101115907200, 469673709120, 1364214519360, 2547438284160, 3056688835200, 2277532857600, 958961203200, 174356582400;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3+12*x+15*x^2+6*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 9 + 42*x + 42*x^2.

%t Mathematica

P[1, x_] := 9 + 42*x + 42*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3 + 12*x + 15*x^2 + 6*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 9 + 42*x + 42*x^2; for(n = 2, N, my(i = n); v[i] = ((3 + 12*x + 15*x^2 + 6*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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, 0]
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)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257180 → x ↦ x+1

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