A257180 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

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

Technical details

Node id: n-e23fac9b1096b64b · revision 0 · JSON
Definition source: derived:shift_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:           42,           42,             9;
   2:          336,          504,           222,            27;
   3:         3024,         6048,          4074,          1050,            81;
   4:        30240,        75600,         68880,         27720,          4686,           243;
   5:       332640,       997920,       1149120,        635040,        171402,         20202,          729;
   6:      3991680,     13970880,      19474560,      13759200,       5181456,        998424,        85182,         2187;
   7:     51891840,    207567360,     339958080,     293388480,     142947504,      39076128,      5588394,       353850,        6561;
   8:    726485760,   3269185920,    6155170560,    6286896000,    3776582880,    1352317680,    279163920,     30418920,     1454766,     19683;
   9:  10897286400,  54486432000,  115978262400,  136994457600,   98124475680,   43735839840,  11978699040,   1918334880,   162277962,   5937162,    59049;
  10: 174356582400, 958961203200, 2277532857600, 3056688835200, 2547438284160, 1364214519360, 469673709120, 101115907200, 12808232976, 852949944, 24102942, 177147;

%F Formula

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

%t Mathematica

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

%o PARI/GP

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

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/(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