A290316 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-29 - 30\,x + 18\,n + 18\,n x\right)\,P_{n-1}(x) + \left(-6 - 15\,x - 9\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	7	6														13
3	139	228	90													457
4	4609	11142	8694	2160												26605
5	214297	684744	798876	399708	71280											2168905
6	12820999	50952270	79197516	59811912	21739428	2993760										227515885
7	937923283	4458531852	8662466646	8765879400	4841567964	1370228400	152681760									29189279305
8	81109986433	448826839710	1046669680170	1329010724880	987810893820	426954893832	98362803600	9160905600								4427906728045
9	8094667157329	51108049132896	139090485294000	212565825877140	198847674546300	116052168719376	40989918282984	7928804037600	632102486400							775309695534025
10	915646445959303	6495808932145206	20209998782523816	36119577412673640	40759169905020240	30012962662763352	14352852961217304	4269416221548864	709401128349600	49303993939200						153894138446140525
11	115769395774250971	911663193081238836	3191871057553772658	6532107338829618000	8635221054951288840	7684726211158603824	4646154310151007024	1875416621597119152	480386942879044320	69778844115494400	4289447472710400					34147384417564148425
12	16178897851297498561	140036724133338488502	544910209112388457278	1256548274464008630384	1904736354517964829960	1988710938727020701424	1455455865750931816128	744017958652702570272	259094112047104927536	58135244533854137280	7485488188617196800	411786957380198400				8375721854936609452525
13	2476453714715956273993	23368550237650341034488	100048578860316829934628	256687350300714477996108	438934487823423806879448	526131718426497237237792	452333709555539942429712	280283658349648992924672	123790228489206857835792	37820760616270608236352	7535158988850911615040	869817131797568371200	43237630524920832000			2250323710125158451601225
14	412037876955756202893511	4209848227774411268784414	19666504615247810213799204	55565830733269639024864584	105816403271062486613457324	143225295518278981263300912	141513783881691565631404656	103123603923107362317184896	55273174138703883387346416	21444651668281787503265376	5825385922381632796724352	1041977270774784998810880	108846725323672505433600	4929089879840974848000		657232272862733614702118125
15	74042038979690777284629667	814316734563424712874129756	4121922461234323455753269166	12717273169800363437613905976	26691532072700373751432904676	40267449103590161796122677872	44966322627429318539364015264	37693261576879446079985783904	23782920775085956494834312624	11209503176797223916616866624	3871039234322396987599592736	945161196712824159027060480	153173976413361111625639680	14594505877197455560243200	606278055220439906304000	207323118928441283115601335625

Technical details

Node id: n-bdb6efcdc16252c4 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-0d53f5dfcd557f8e

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
   1:               1;
   2:               7,                6;
   3:             139,              228,                90;
   4:            4609,            11142,              8694,              2160;
   5:          214297,           684744,            798876,            399708,             71280;
   6:        12820999,         50952270,          79197516,          59811912,          21739428,           2993760;
   7:       937923283,       4458531852,        8662466646,        8765879400,        4841567964,        1370228400,         152681760;
   8:     81109986433,     448826839710,     1046669680170,     1329010724880,      987810893820,      426954893832,       98362803600,       9160905600;
   9:   8094667157329,   51108049132896,   139090485294000,   212565825877140,   198847674546300,   116052168719376,    40989918282984,    7928804037600,    632102486400;
  10: 915646445959303, 6495808932145206, 20209998782523816, 36119577412673640, 40759169905020240, 30012962662763352, 14352852961217304, 4269416221548864, 709401128349600, 49303993939200;

%F Formula

The row polynomials satisfy P(n,x) = (18*n-29+(18*n-30)*x)*P(n-1,x) - (6+15*x+9*x^2)*d/dx 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[(18*n - 29 + (18*n - 30)*x)*P[n - 1, x] - (6 + 15*x + 9*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] = 1; for(n = 2, N, my(i = n); v[i] = (18*n - 29 + (18*n - 30)*x)*v[i - 1] - (6 + 15*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, 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 T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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