A111006 → basis A013610

nonnegative coefficients

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

Recurrence

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

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

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

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	3	15	18													36
3	5	39	99	81												224
4	11	111	414	675	405											1616
5	21	267	1341	3321	4050	1944										10944
6	43	651	4068	13419	24624	23814	9477									76096
7	85	1503	11295	46737	114939	167913	134865	45927								523264
8	171	3447	30186	149931	461781	902745	1093500	750141	223074							3614976
9	341	7731	77409	449145	1663740	4078998	6617133	6847497	4100625	1082565						24925184
10	683	17187	193536	1283931	5555790	16381116	33321861	46165383	41682033	22143375	5255361					172000256
11	1365	37767	472563	3529089	17474049	60222447	147384117	256091139	309561102	247887702	118334196	25509168				1186504704
12	2731	82383	1133766	9411147	52470099	206964315	592127334	1237901049	1876583781	2011484502	1446936696	627100380	123825753			8186023936
13	5461	178395	2678229	24463593	151666506	673847748	2207056977	5394924405	9840686436	13227881418	12735097830	8314925886	3299717169	601059771		56474189824
14	10923	384123	6246972	62265915	424901700	2099707110	7747764615	21683722275	46248882660	74803273200	90305230023	78900210918	47158480017	17258546475	2917611090	389617238016

Technical details

Node id: n-53f3ce2888a3b03b · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-ac4ba740a33f010b

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:   3,   15,    18;
  3:   5,   39,    99,     81;
  4:  11,  111,   414,    675,     405;
  5:  21,  267,  1341,   3321,    4050,    1944;
  6:  43,  651,  4068,  13419,   24624,   23814,    9477;
  7:  85, 1503, 11295,  46737,  114939,  167913,  134865,   45927;
  8: 171, 3447, 30186, 149931,  461781,  902745, 1093500,  750141,  223074;
  9: 341, 7731, 77409, 449145, 1663740, 4078998, 6617133, 6847497, 4100625, 1082565;

%F Formula

The row polynomials satisfy P(n,x) = (2+9*x+9*x^2)*P(n-2,x) + (1+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[(2 + 9*x + 9*x^2)*P[n - 2, x] + (1 + 3*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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 9*x + 9*x^2)*v[i - 2] + (1 + 3*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 Another version of Fibonacci-Pascal triangle A037027.?
  - 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