A257619 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-5 - 7\,x + 9\,n + 9\,n x\right)\,P_{n-1}(x) + \left(-9\,x - 9\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

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

Technical details

Node id: n-9d3de644c7283a2d · revision 0 · JSON
Definition source: derived:shift_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:              4,              2;
  2:             52,             52,               4;
  3:           1144,           1716,             588,               8;
  4:          35464,          70928,           41964,            6500,             16;
  5:        1418560,        3546400,         2980120,          923780,          71564,             32;
  6:       69509440,      208528320,       227146920,       106746640,       19405932,         787332,            64;
  7:     4031547520,    14110416320,     18886707840,     11940728800,     3522583064,      398353956,       8660908,          128;
  8:   270113683840,  1080454735360,   1715631998080,   1365304420480,   562947769384,   110918695888,    8079670924,     95270500,        256;
  9: 20528639971840, 92378879873280, 169736213687040, 162975308496000, 86687218537920, 24830494809120, 3394394941080, 162831934980, 1047976524, 512;

%F Formula

The row polynomials satisfy P(n,x) = (9*n-5+(9*n-7)*x)*P(n-1,x) - (9*x+9*x^2)*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[(9*n - 5 + (9*n - 7)*x)*P[n - 1, x] - (9*x + 9*x^2)*D[P[n - 1, x], x]];
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] = (9*n - 5 + (9*n - 7)*x)*v[i - 1] - (9*x + 9*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 - 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]
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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
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); 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/(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