A257619 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x) + \left(9\,x + 27\,x^{2} + 27\,x^{3} + 9\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	4	52	52													108
3	8	588	1716	1144												3456
4	16	6500	41964	70928	35464											154872
5	32	71564	923780	2980120	3546400	1418560										8940456
6	64	787332	19405932	106746640	227146920	208528320	69509440									632124648
7	128	8660908	398353956	3522583064	11940728800	18886707840	14110416320	4031547520								52898998536
8	256	95270500	8079670924	110918695888	562947769384	1365304420480	1715631998080	1080454735360	270113683840							5113546244712
9	512	1047976524	162831934980	3394394941080	24830494809120	86687218537920	162975308496000	169736213687040	92378879873280	20528639971840						560695030228296
10	1024	11527743812	3270262394412	102019755860880	1048785045920040	5067519063647040	13374290984134080	20485421782425600	18208363427654400	8724671988032000	1744934397606400					68759288235419688
11	2048	126805186028	65555108557796	3030518692642584	43016444176648800	280124797828212480	997268729230213440	2107261292937440640	2719942153068940800	2107983395508390400	902131083562508800	164023833375001600				9324847930293745416
12	4096	1394857054500	1312750638574284	89327254474906448	1728570958184574504	14886523264991938560	69573163930474364160	194813570336286021120	342462225953910142080	381678225460435904000	262174648305963993600	101366729025750988800	16894454837625164800			1385668753473593630952
13	8192	15343427615884	26273145913194180	2619370893820921240	68454841299735931040	768809431782004092480	4625536820091169382400	16698125579826104494080	38394713933120513928960	57706421885713350830080	56562716808285739251200	34882456572361465267200	12299163121791119974400	1892178941814018457600		223901223595468383348936
14	16384	168777703807492	525662382822890092	76539765130896987920	2682484467098413937960	38872236945743629588160	296701724082423685945920	1353659298253984616294400	3959983244899267031846400	7707631275431367177497600	10108815806647922831667200	8840023493787367481600000	4945800970350254119705600	1602675563716473633587200	228953651959496233369600	39085876816137690278741928

Technical details

Node id: n-8dfc00cbda2ca4e8 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-cc3d28c31b2705bc

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 0)
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:   2,          4;
  2:   4,         52,           52;
  3:   8,        588,         1716,          1144;
  4:  16,       6500,        41964,         70928,          35464;
  5:  32,      71564,       923780,       2980120,        3546400,        1418560;
  6:  64,     787332,     19405932,     106746640,      227146920,      208528320,        69509440;
  7: 128,    8660908,    398353956,    3522583064,    11940728800,    18886707840,     14110416320,      4031547520;
  8: 256,   95270500,   8079670924,  110918695888,   562947769384,  1365304420480,   1715631998080,   1080454735360,   270113683840;
  9: 512, 1047976524, 162831934980, 3394394941080, 24830494809120, 86687218537920, 162975308496000, 169736213687040, 92378879873280, 20528639971840;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2+8*x+10*x^2+4*x^3)*P(n-1,x) + (9*x+27*x^2+27*x^3+9*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2 + 8*x + 10*x^2 + 4*x^3)*P[n - 1, x] + (9*x + 27*x^2 + 27*x^3 + 9*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2 + 8*x + 10*x^2 + 4*x^3)*v[i - 1] + (9*x + 27*x^2 + 27*x^3 + 9*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 - 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 [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 2.?
  - 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257619 → 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