A131222 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 + 6\,n - 2\,n^{2}\right)\,P_{n-2}(x) + \left(-2 + 4\,x + 3\,n\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	4	20	16													40
3	24	140	192	64												420
4	192	1256	2288	1408	256											5400
5	1920	13736	30160	25920	8960	1024										81720
6	23040	177216	445984	479040	236800	52224	4096									1418400
7	322560	2635104	7372960	9330496	5877760	1878016	286720	16384								27720000
8	5160960	44376384	135282880	194523392	146741504	60440576	13475840	1507328	65536							601574400
9	92897280	834921792	2733806016	4359200768	3788322048	1887642624	546545664	89751552	7667712	262144						14341017600
10	1857945600	17359200000	60405520896	104981477120	102379043840	59303838720	20913328128	4482170880	564264960	38010880	1048576					372285849600
11	40874803200	395281059840	1449922864896	2711391736064	2911778298880	1908159503360	787150307328	206445330432	34040709120	3388211200	184549376	4194304				10448621568000
12	980995276800	9784031247360	37589287048704	74891085519360	87322639461376	63477691791360	29794816274432	9181664870400	1859027140608	242999623680	19596836864	880803840	16777216			315144732672000
13	25505877196800	261578937461760	1047160109458944	2205519894090240	2761792531244032	2194211045238784	1148511288184832	404399298068480	96523916476416	15532606881792	1648360423424	109869793280	4143972352	67108864		10162497945600000
14	714164561510400	7512563258081280	31204862569009152	69043417519405056	92049117314172928	79010572095623168	45421313030725632	17905337635913728	4898537383460864	931584058589184	121950631362560	10713422626816	600003575808	19260243968	268435456	348824753012736000

Technical details

Node id: n-cb6978de4b461a84 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-d5b9542b953ee0df

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:        1;
  1:        1,         4;
  2:        4,        20,         16;
  3:       24,       140,        192,         64;
  4:      192,      1256,       2288,       1408,        256;
  5:     1920,     13736,      30160,      25920,       8960,       1024;
  6:    23040,    177216,     445984,     479040,     236800,      52224,      4096;
  7:   322560,   2635104,    7372960,    9330496,    5877760,    1878016,    286720,    16384;
  8:  5160960,  44376384,  135282880,  194523392,  146741504,   60440576,  13475840,  1507328,   65536;
  9: 92897280, 834921792, 2733806016, 4359200768, 3788322048, 1887642624, 546545664, 89751552, 7667712, 262144;

%F Formula

The row polynomials satisfy P(n,x) = -(2*n^2-6*n+4)*P(n-2,x) + (3*n-2+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(2*n^2 - 6*n + 4)*P[n - 2, x] + (3*n - 2 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(2*n^2 - 6*n + 4)*v[i - 2] + (3*n - 2 + 4*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 - 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 ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Exponential Riordan array [1, log((1-x)/(1-2x))].?
  - 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