A269951 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	2														3
3	3	8	4													15
4	11	36	32	8												87
5	49	186	224	104	16											579
6	261	1106	1620	1048	304	32										4371
7	1631	7520	12656	10160	4160	832	64									37023
8	11743	57892	108188	101352	52080	14816	2176	128								348375
9	95901	499394	1013540	1067584	646464	232512	48832	5504	256							3609987
10	876809	4780498	10376872	11993456	8235248	3514336	938112	151808	13568	512						40881219
11	8877691	50343944	115598268	144141088	109402928	53268096	16957248	3502080	450816	32768	1024					502575951
12	98641011	578858228	1394475256	1852948488	1527000080	826393984	301580160	74573184	12291840	1290240	77824	2048				6668132343
13	1193556233	7218999546	18129912104	25435869976	22462868736	13264440992	5403757568	1537590912	304432128	41029120	3581952	182272	4096			94996225635
14	15624736141	97085665602	252917540732	371979570504	348554901376	221537577536	98864106176	31513482880	7202021376	1169079296	131349504	9693184	421888	8192		1446590154387
15	220048367319	1400906461760	3770375496936	5780298199312	5702728236816	3861000173888	1861191656896	651979364608	167404100864	31447882752	4265417728	405909504	25665536	966656	16384	23452077916959

Technical details

Node id: n-5362cadbf99c9107 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-60cb00b823fa13b8

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:      1,       2;
   3:      3,       8,        4;
   4:     11,      36,       32,        8;
   5:     49,     186,      224,      104,      16;
   6:    261,    1106,     1620,     1048,     304,      32;
   7:   1631,    7520,    12656,    10160,    4160,     832,     64;
   8:  11743,   57892,   108188,   101352,   52080,   14816,   2176,    128;
   9:  95901,  499394,  1013540,  1067584,  646464,  232512,  48832,   5504,   256;
  10: 876809, 4780498, 10376872, 11993456, 8235248, 3514336, 938112, 151808, 13568, 512;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n - 3)*P[n - 2, x] + (n + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = -(n - 3)*v[i - 2] + (n + 2*x)*v[i - 1]); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[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