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	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

All 15 rows

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.