A115951 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-4 - 12\,x + 4\,n + 12\,n x\right)\,P_{n-2}(x) + \left(-2 - 6\,x + 4\,n + 12\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	6														8
2	8	42	54													104
3	32	252	648	540												1472
4	136	1416	5454	9180	5670											21856
5	592	7680	39420	99900	124740	61236										333568
6	2624	40752	261360	885060	1666980	1653372	673596									5183744
7	11776	213024	1639008	6947640	17508960	26209008	21555072	7505784								81590272
8	53344	1101504	9885456	50334480	158947110	318549672	395400852	277714008	84440070							1296426496
9	243392	5648832	57928608	344373120	1307286540	3284760276	5460169176	5786959464	3546482940	956987460						20750839808
10	1116928	28781760	332009280	2256971040	10009364400	30248563212	63058689540	89506474200	82751268600	44978410620	10909657044					334081306624
11	5149696	145884288	1869626880	14304755520	72580717440	256353124104	642941993232	1144669588920	1417242134880	1161782776440	567302166288	124965162504				5404163080192
12	23835904	736263936	10378539072	88263267840	504258456240	2038379633136	5976621291564	12803569003368	19884454324020	21826969987680	16069924825812	7123014262728	1437099368796			87763693060096
13	110690816	3702506496	56931997824	532753752960	3384320940000	15408794104704	51720313468824	129553802399736	242124469998840	333374237590680	328620689479368	219563790519528	89100160865352	16581915793800		1430025994108928
14	515483648	18562242048	309191081472	3157173452160	22074996303360	111785352430752	422706650157792	1212338187228168	2649556125576360	4390239305972520	5428056223614024	4855021528442904	2969047295932536	1110988358184600	191876454185400	23367175920287744

Technical details

Node id: n-2a67a442e762ff53 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-23e3cf183ada7515

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:      2,       6;
  2:      8,      42,       54;
  3:     32,     252,      648,       540;
  4:    136,    1416,     5454,      9180,       5670;
  5:    592,    7680,    39420,     99900,     124740,      61236;
  6:   2624,   40752,   261360,    885060,    1666980,    1653372,     673596;
  7:  11776,  213024,  1639008,   6947640,   17508960,   26209008,   21555072,    7505784;
  8:  53344, 1101504,  9885456,  50334480,  158947110,  318549672,  395400852,  277714008,   84440070;
  9: 243392, 5648832, 57928608, 344373120, 1307286540, 3284760276, 5460169176, 5786959464, 3546482940, 956987460;

%F Formula

The row polynomials satisfy n*P(n,x) = (4*n-4+(12*n-12)*x)*P(n-2,x) + (4*n-2+(12*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (12*n - 12)*x)*P[n - 2, x] + (4*n - 2 + (12*n - 6)*x)*P[n - 1, x])/(n)]];
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] = 2 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (12*n - 12)*x)*v[i - 2] + (4*n - 2 + (12*n - 6)*x)*v[i - 1])/(n)); 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 Expansion of 1/sqrt(1-4*x*y-4*x^2*y).?
  - 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