A107131 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-3 - 8\,x + 16\,x^{2} + 3\,n + 8\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(1 + 4\,x + 2\,n + 8\,n x\right)\,P_{n-1}(x)\)

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

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

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	2	12	16													30
3	4	36	96	64												200
4	9	104	416	640	256											1425
5	21	300	1600	3840	3840	1024										10625
6	51	864	5760	18880	30720	21504	4096									81875
7	127	2492	19936	82880	188160	222208	114688	16384								646875
8	323	7200	67200	338688	989184	1662976	1490944	589824	65536							5211875
9	835	20844	222336	1317120	4709376	10321920	13418496	9437184	2949120	262144						42659375
10	2188	60460	725520	4940160	20912640	56641536	97198080	100761600	57016320	14417920	1048576					353725000
11	5798	175692	2342560	18025920	88197120	284798976	609337344	843448320	713687040	331612160	69206016	4194304				2965031250
12	15511	511416	7500768	64366720	357477120	1341407232	3444621312	5994971136	6844907520	4815585280	1868562432	327155712	16777216			25083859375
13	41835	1490996	23855936	225871360	1403953408	6005615616	18014650368	37870829568	54796025856	52518715392	31188844544	10250878976	1526726656	67108864		213894609375
14	113634	4353132	75454288	781347840	5374457088	25820354560	88660475904	218811580416	384753008640	470756622336	384162594816	195039330304	54962159616	7046430720	268435456	1836516718750

Technical details

Node id: n-b9d085565acadb7d · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-bdcd1d522b684b55

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:   2,    12,     16;
  3:   4,    36,     96,      64;
  4:   9,   104,    416,     640,     256;
  5:  21,   300,   1600,    3840,    3840,     1024;
  6:  51,   864,   5760,   18880,   30720,    21504,     4096;
  7: 127,  2492,  19936,   82880,  188160,   222208,   114688,   16384;
  8: 323,  7200,  67200,  338688,  989184,  1662976,  1490944,  589824,   65536;
  9: 835, 20844, 222336, 1317120, 4709376, 10321920, 13418496, 9437184, 2949120, 262144;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = (3*n-3+(8*n-8)*x-(16*n-16)*x^2)*P(n-2,x) + (2*n+1+(8*n+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[Together[((3*n - 3 + (8*n - 8)*x - (16*n - 16)*x^2)*P[n - 2, x] + (2*n + 1 + (8*n + 4)*x)*P[n - 1, x])/(n + 2)]];
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] = ((3*n - 3 + (8*n - 8)*x - (16*n - 16)*x^2)*v[i - 2] + (2*n + 1 + (8*n + 4)*x)*v[i - 1])/(n + 2)); 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)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from A Motzkin related triangle.?
  - 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