A225466 → reverse

nonnegative coefficients

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

Recurrence

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

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

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	9	21	4													34
4	27	135	117	8												287
5	81	702	1431	609	16											2839
6	243	3240	12015	13275	3093	32										31898
7	729	13851	81405	171990	115479	15561	64									399079
8	2187	56133	479682	1655640	2238327	970515	77997	128								5480609
9	6561	218700	2561706	13121514	29893941	27533142	7998111	390369	256							81724300
10	19683	826686	12715218	90598662	312747561	501114600	326858895	65155995	1952613	512						1311990425
11	59049	3050865	59639490	564246000	2750215923	6820052337	7996181085	3790915830	527105799	9764601	1024					22521232003
12	177147	11042163	267393555	3243364740	21228305769	75464475471	139929432984	123319282680	43281391527	4246140195	48826077	2048				411039834356
13	531441	39326634	1155529881	17484507315	148012400547	714644459100	1929077808372	2748758208768	1856314132101	488833727382	34115599791	244136529	4096			7938680371957
14	1594323	138174660	4843021833	89430478137	951087913776	5992255791522	22224055984416	46827830793744	52297831945359	27454899031560	5479517800575	273657207915	1220694933	8192		161596770440945
15	4782969	479891223	19779702579	437797198566	5715039041712	45558316874070	222470818532820	651636780022800	1093450111710957	971427840165783	400807139843565	61095667430070	2192919748119	6103499241	16384	3454818798460858

Technical details

Node id: n-6d4cddeff104eab5 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-168559fd18c897bc

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:     9,     21,        4;
   4:    27,    135,      117,        8;
   5:    81,    702,     1431,      609,        16;
   6:   243,   3240,    12015,    13275,      3093,        32;
   7:   729,  13851,    81405,   171990,    115479,     15561,        64;
   8:  2187,  56133,   479682,  1655640,   2238327,    970515,     77997,      128;
   9:  6561, 218700,  2561706, 13121514,  29893941,  27533142,   7998111,   390369,     256;
  10: 19683, 826686, 12715218, 90598662, 312747561, 501114600, 326858895, 65155995, 1952613, 512;

%F Formula

The row polynomials satisfy P(n,x) = (3+(3*n-4)*x)*P(n-1,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[(3 + (3*n - 4)*x)*P[n - 1, 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] = (3 + (3*n - 4)*x)*v[i - 1] - 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 read by rows, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >=…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A225466: Triangle read by rows, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >…

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.