A290595 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 4\,x + 6\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-3\,x - 3\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	2														5
3	27	27	4													58
4	405	540	171	8												1124
5	8505	14175	6885	951	16											30532
6	229635	459270	300510	72198	4995	32										1066640
7	7577955	17681895	14543550	5038362	682479	25551	64									45549856
8	295540245	788107320	780529365	354760560	73848915	6073056	129099	128								2298988688
9	13299311025	39897933075	46202791635	26040379365	7403662035	994529745	52070121	648567	256							133891325824
10	678264862275	2260882874250	2996778084300	2017751766030	731429268570	139454339310	12658141188	436017978	3249747	512						8837658604160
11	38661097149675	141757356215475	211618637029800	165777980708340	73269352406250	18291379816710	2446028256672	154936200276	3595385475	16264095	1024					651980379433792
12	2435649120429525	9742596481718100	16173225640947375	14454095521025160	7549630529010570	2343373279327080	421024728434310	40751716005000	1844518236561	29348591220	81354267	2048				53162220965081216
13	168059789309637225	728259087008427975	1330676302794663825	1336269274848983475	806135423700256650	300183659250905430	68339726736691410	9113502751623270	653527344715245	21522341433411	237961546173	406845063	4096			4747712054455733248
14	12604484198222791875	58820926258373028750	117305732938126783050	130775413829462151750	89538340663736689725	38970540571260998700	10799588586762235260	1858923683321484780	188259441358073085	10182455214817158	247454025345306	1920779862030	2034385059	8192		460872642002678654720
15	1020963220056046141875	5104816100280230709375	11033125168177683821250	13521530448555712332750	10362958748340620447025	5168667556951589916225	1693608135818806787580	359925958494820497780	47909261823839861445	3750263084687368977	155174528300050818	2814191712175806	15457786223895	10172269359	16384	48317412865772008620544

Technical details

Node id: n-c2202905f6c6d864 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-29d82cb73104e710

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:            3,             2;
   3:           27,            27,             4;
   4:          405,           540,           171,             8;
   5:         8505,         14175,          6885,           951,           16;
   6:       229635,        459270,        300510,         72198,         4995,           32;
   7:      7577955,      17681895,      14543550,       5038362,       682479,        25551,          64;
   8:    295540245,     788107320,     780529365,     354760560,     73848915,      6073056,      129099,       128;
   9:  13299311025,   39897933075,   46202791635,   26040379365,   7403662035,    994529745,    52070121,    648567,     256;
  10: 678264862275, 2260882874250, 2996778084300, 2017751766030, 731429268570, 139454339310, 12658141188, 436017978, 3249747, 512;

%F Formula

The row polynomials satisfy P(n,x) = (6*n-9+(3*n-4)*x)*P(n-1,x) - (3*x+3*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[(6*n - 9 + (3*n - 4)*x)*P[n - 1, x] - (3*x + 3*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] = (6*n - 9 + (3*n - 4)*x)*v[i - 1] - (3*x + 3*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