A029635 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 3 + 4\,x\)

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	4														7
2	6	14	8													28
3	12	40	44	16												112
4	24	104	168	120	32											448
5	48	256	544	576	304	64										1792
6	96	608	1600	2240	1760	736	128									7168
7	192	1408	4416	7680	8000	4992	1728	256								28672
8	384	3200	11648	24192	31360	25984	13440	3968	512							114688
9	768	7168	29696	71680	111104	114688	78848	34816	8960	1024						458752
10	1536	15872	73728	202752	365568	451584	387072	227328	87552	19968	2048					1835008
11	3072	34816	179200	552960	1136640	1634304	1677312	1228800	629760	215040	44032	4096				7340032
12	6144	75776	428032	1464320	3379200	5541888	6623232	5812224	3717120	1689600	518144	96256	8192			29360128
13	12288	163840	1007616	3784704	9687040	17842176	24330240	24870912	19058688	10813440	4415488	1228800	208896	16384		117440512
14	24576	352256	2342912	9584640	26943488	55058432	84344832	98402304	87859200	59744256	30457856	11288576	2875392	450560	32768	469762048

Technical details

Node id: n-f124fdca65bac486 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-59885d546c14be4d

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,    4;
  2:   6,   14,     8;
  3:  12,   40,    44,    16;
  4:  24,  104,   168,   120,     32;
  5:  48,  256,   544,   576,    304,     64;
  6:  96,  608,  1600,  2240,   1760,    736,   128;
  7: 192, 1408,  4416,  7680,   8000,   4992,  1728,   256;
  8: 384, 3200, 11648, 24192,  31360,  25984, 13440,  3968,  512;
  9: 768, 7168, 29696, 71680, 111104, 114688, 78848, 34816, 8960, 1024;

%F Formula

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

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (2*n - 2)*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (2*n - 2)*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 ?

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.