A095660 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 3\)

\(P_{1}(x) = 4 + 9\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	3															3
1	4	9														13
2	8	30	27													65
3	16	84	144	81												325
4	32	216	540	594	243											1625
5	64	528	1728	2808	2268	729										8125
6	128	1248	5040	10800	12960	8262	2187									40625
7	256	2880	13824	36720	58320	55404	29160	6561								203125
8	512	6528	36288	114912	226800	285768	224532	100602	19683							1015625
9	1024	14592	92160	338688	798336	1251936	1306368	874800	341172	59049						5078125
10	2048	32256	228096	953856	2612736	4898880	6368544	5668704	3306744	1141614	177147					25390625
11	4096	70656	552960	2592000	8087040	17635968	27433728	30443040	23619600	12203460	3779136	531441				126953125
12	8192	153600	1317888	6842880	23950080	59533056	107775360	143187264	138568320	95265720	44168652	12400290	1594323			634765625
13	16384	331776	3096576	17639424	68428800	190916352	394149888	609700608	706698432	606236400	374134464	157306536	40389516	4782969		3173828125
14	32768	712704	7188480	44568576	189775872	587119104	1361048832	2401850880	3242498688	3332568096	2566978128	1437016464	552698640	130734486	14348907	15869140625

Technical details

Node id: n-c6582f5662aecad5 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-dc0f84d4ff37664b

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    3;
  1:    4,     9;
  2:    8,    30,    27;
  3:   16,    84,   144,     81;
  4:   32,   216,   540,    594,    243;
  5:   64,   528,  1728,   2808,   2268,     729;
  6:  128,  1248,  5040,  10800,  12960,    8262,    2187;
  7:  256,  2880, 13824,  36720,  58320,   55404,   29160,   6561;
  8:  512,  6528, 36288, 114912, 226800,  285768,  224532, 100602,  19683;
  9: 1024, 14592, 92160, 338688, 798336, 1251936, 1306368, 874800, 341172, 59049;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(3*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 3, P(1,x) = 4 + 9*x.

%t Mathematica

P[0, x_] := 3;
P[1, x_] := 4 + 9*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (3*n - 3)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 3; v[2] = 4 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (3*n - 3)*x)*v[i - 1])/(n - 1)); 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 (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Pascal (1,3) triangle.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - 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

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.