A029635 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 2\)

\(P_{1}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	3	2														5
2	6	7	2													15
3	12	20	11	2												45
4	24	52	42	15	2											135
5	48	128	136	72	19	2										405
6	96	304	400	280	110	23	2									1215
7	192	704	1104	960	500	156	27	2								3645
8	384	1600	2912	3024	1960	812	210	31	2							10935
9	768	3584	7424	8960	6944	3584	1232	272	35	2						32805
10	1536	7936	18432	25344	22848	14112	6048	1776	342	39	2					98415
11	3072	17408	44800	69120	71040	51072	26208	9600	2460	420	43	2				295245
12	6144	37888	107008	183040	211200	173184	103488	45408	14520	3300	506	47	2			885735
13	12288	81920	251904	473088	605440	557568	380160	194304	74448	21120	4312	600	51	2		2657205
14	24576	176128	585728	1198080	1683968	1720576	1317888	768768	343200	116688	29744	5512	702	55	2	7971615

Technical details

Node id: n-5896c729e0a05938 · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-70d79adb4c85c3c6

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:   2;
  1:   3,    2;
  2:   6,    7,    2;
  3:  12,   20,   11,    2;
  4:  24,   52,   42,   15,    2;
  5:  48,  128,  136,   72,   19,    2;
  6:  96,  304,  400,  280,  110,   23,    2;
  7: 192,  704, 1104,  960,  500,  156,   27,   2;
  8: 384, 1600, 2912, 3024, 1960,  812,  210,  31,  2;
  9: 768, 3584, 7424, 8960, 6944, 3584, 1232, 272, 35, 2;

%F Formula

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

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*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] = 2; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*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 The (1,2)-Pascal triangle (or Lucas triangle) read by rows.?
  - 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

Related ?

Same fingerprint ?

A136157 → basis A013609

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.