A107131 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-3 - 6\,x + 9\,x^{2} + 3\,n + 6\,n x - 9\,n x^{2}\right)\,P_{n-2}(x) + \left(1 + 3\,x + 2\,n + 6\,n x\right)\,P_{n-1}(x)\)

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

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

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	3														4
2	2	9	9													20
3	4	27	54	27												112
4	9	78	234	270	81											672
5	21	225	900	1620	1215	243										4224
6	51	648	3240	7965	9720	5103	729									27456
7	127	1869	11214	34965	59535	52731	20412	2187								183040
8	323	5400	37800	142884	312984	394632	265356	78732	6561							1244672
9	835	15633	125064	555660	1490076	2449440	2388204	1259712	295245	19683						8599552
10	2188	45345	408105	2084130	6616890	13441302	17299170	13450050	5708070	1082565	59049					60196864
11	5798	131769	1317690	7604685	27906120	67584132	108448956	112586760	71449290	24898995	3897234	177147				426008576
12	15511	383562	4219182	27154710	113107995	318322224	613068588	800232048	685263645	361576710	105225318	13817466	531441			3042918400
13	41835	1118247	13418964	95289480	444219633	1425160737	3206220732	5055145524	5485789881	3943351269	1756353456	432947268	48361131	1594323		21909012480
14	113634	3264849	42443037	329631120	1700511813	6127291170	15779659896	29207820213	38518745265	35346613302	21633545934	8237512647	1741000716	167403915	4782969	158840340480

Technical details

Node id: n-47a2e536b5967de9 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-0a40c69c4c7d8789

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,     3;
  2:   2,     9,      9;
  3:   4,    27,     54,     27;
  4:   9,    78,    234,    270,      81;
  5:  21,   225,    900,   1620,    1215,     243;
  6:  51,   648,   3240,   7965,    9720,    5103,     729;
  7: 127,  1869,  11214,  34965,   59535,   52731,   20412,    2187;
  8: 323,  5400,  37800, 142884,  312984,  394632,  265356,   78732,   6561;
  9: 835, 15633, 125064, 555660, 1490076, 2449440, 2388204, 1259712, 295245, 19683;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = (3*n-3+(6*n-6)*x-(9*n-9)*x^2)*P(n-2,x) + (2*n+1+(6*n+3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (6*n - 6)*x - (9*n - 9)*x^2)*P[n - 2, x] + (2*n + 1 + (6*n + 3)*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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (6*n - 6)*x - (9*n - 9)*x^2)*v[i - 2] + (2*n + 1 + (6*n + 3)*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