A290316 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 - 5\,x + 9\,n + 3\,n x\right)\,P_{n-1}(x) + \left(9\,x - 3\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	6	1														7
3	90	48	1													139
4	2160	2214	234	1												4609
5	71280	114588	27432	996	1											214297
6	2993760	6770628	2791800	260748	4062	1										12820999
7	152681760	454137840	280652364	48256344	2178630	16344	1									937923283
8	9160905600	34236464400	29157089832	7846782660	691711920	16966530	65490	1								81109986433
9	632102486400	2871984146400	3187159638984	1219804572672	174637926180	8851303620	126820980	262092	1							8094667157329
10	49303993939200	265665182896800	369150976563264	188635513271256	39425596747272	3359003385600	105253405560	924701832	1048518	1						915646445959303
11	4289447472710400	26884369388390400	45402482111562720	29625769997314992	8433423152068320	1074197994855552	58507774196760	1191239330400	6639627186	4194240	1					115769395774250971
12	411786957380198400	2955831657435014400	5928645303593081280	4779031762458812016	1764978796013948064	311696957163930912	25963334047202832	950008118091480	13027850671248	47219770782	16777146	1				16178897851297498561
13	43237630524920832000	350965565498518387200	820854153722434443840	796834261145169461952	368337804299436977424	85584077086896494208	10050412299619937616	575014708594613952	14656035104000136	139061384077932	333809938944	67108788	1			2476453714715956273993
14	4929089879840974848000	44768556885739832409600	120285577518310971751680	137800109586864361894272	77562029956980291548256	22816313364302467014672	3571318962247872392064	292735437114025484208	11926257866857848528	217645426247198508	1458711068323896	2350963744452	268435374	1		412037876955756202893511
15	606278055220439906304000	6106613104111296871987200	18616703034854220816142080	24759726073652227323697920	16604166160352661574895136	6004877483998262621816064	1202531290425435088762128	133017808472661932688096	7887593589203764662912	235219792824154072080	3140095998766570452	15109024634507304	16518754782558	1073741736	1	74042038979690777284629667

Technical details

Node id: n-ce3ecfa80ae30839 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-a228985e3530ab5e

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:              6,               1;
   3:             90,              48,               1;
   4:           2160,            2214,             234,               1;
   5:          71280,          114588,           27432,             996,              1;
   6:        2993760,         6770628,         2791800,          260748,           4062,             1;
   7:      152681760,       454137840,       280652364,        48256344,        2178630,         16344,            1;
   8:     9160905600,     34236464400,     29157089832,      7846782660,      691711920,      16966530,        65490,         1;
   9:   632102486400,   2871984146400,   3187159638984,   1219804572672,   174637926180,    8851303620,    126820980,    262092,       1;
  10: 49303993939200, 265665182896800, 369150976563264, 188635513271256, 39425596747272, 3359003385600, 105253405560, 924701832, 1048518, 1;

%F Formula

The row polynomials satisfy P(n,x) = (9*n-12+(3*n-5)*x)*P(n-1,x) + (9*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[(9*n - 12 + (3*n - 5)*x)*P[n - 1, x] + (9*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] = (9*n - 12 + (3*n - 5)*x)*v[i - 1] + (9*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 (−∞, 0]
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)

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

Related ?

Same fingerprint ?

A290316: Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the …

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