A056241 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 9\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 6\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	3														5
2	5	15	9													29
3	14	63	81	27												185
4	41	246	495	378	81											1241
5	122	915	2520	3105	1620	243										8525
6	365	3285	11502	19737	17010	6561	729									59189
7	1094	11487	48825	107730	130977	85050	25515	2187								412865
8	3281	39372	196866	531576	838107	775656	398034	96228	6561							2885681
9	9842	132867	763992	2441502	4739958	5727753	4225284	1771470	354294	19683						20186645
10	29525	442875	2878695	10629090	24538545	36696645	35549685	21585690	7577955	1279395	59049					141267149
11	88574	1461471	10595673	44392293	118859400	212201451	255493359	204749127	104792292	31394385	4546773	177147				988751945
12	265721	4782978	38263833	179361810	546605658	1135363554	1637416377	1636982622	1110711690	488020302	126660105	15943230	531441			6920909321
13	797162	15544659	136015776	705339063	2411123598	5715218574	9615819096	11548928079	9808183242	5736295422	2195441820	499731687	55269864	1594323		48445302365
14	2391485	50221185	477101268	2711944179	10278081009	27387818010	52704615789	74024659647	75724575381	55600971426	28430971569	9591624315	1934445240	189724437	4782969	339113927909

Technical details

Node id: n-dc8f9b0d78276a3e · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-e4c9f9bbfb759620

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,      3;
  2:    5,     15,      9;
  3:   14,     63,     81,      27;
  4:   41,    246,    495,     378,      81;
  5:  122,    915,   2520,    3105,    1620,     243;
  6:  365,   3285,  11502,   19737,   17010,    6561,     729;
  7: 1094,  11487,  48825,  107730,  130977,   85050,   25515,    2187;
  8: 3281,  39372, 196866,  531576,  838107,  775656,  398034,   96228,   6561;
  9: 9842, 132867, 763992, 2441502, 4739958, 5727753, 4225284, 1771470, 354294, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(3+9*x+9*x^2)*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(3 + 9*x + 9*x^2)*P[n - 2, x] + (4 + 6*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] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(3 + 9*x + 9*x^2)*v[i - 2] + (4 + 6*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 Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=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
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

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