A049403 → basis A013610

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 4 + 15\,x + 9\,x^{2}\)

\(P_{1}(x) = 10 + 54\,x + 81\,x^{2} + 27\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	4	15	9															28
1	10	54	81	27														172
2	26	192	459	378	81													1136
3	76	690	2250	3105	1620	243												7984
4	232	2538	10530	20385	18225	6561	729											59200
5	764	9660	48384	120960	155925	96957	25515	2187										460352
6	2620	37824	222516	681912	1153845	1054620	479682	96228	6561									3735808
7	9496	153252	1032264	3744468	7868826	9598743	6491016	2243862	354294	19683								31515904
8	35696	638580	4851900	20335860	51098040	78366771	71722665	37069650	10038330	1279395	59049							275495936
9	140152	2744808	23179860	110145420	322230150	596918322	694982673	491123655	199191960	43302600	4546773	177147						2488683520
10	568504	12113280	112625964	598383720	1997933850	4343539464	6167613375	5602923414	3127530285	1017611100	181221381	15943230	531441					23178539008
11	2390480	54967224	557351496	3272166612	12277748340	30664255050	51512821074	57594695301	41679618552	18731622195	4981964130	739234431	55269864	1594323				222125699072
12	10349536	255601752	2807774424	18051883668	75197438316	212220274266	412280830962	549613848069	493571911833	289373305221	106453596249	23519630043	2950028991	189724437	4782969			2186500980736
13	46206736	1218730320	14408377200	100638523440	460766811960	1450275090264	3201525447120	4968453718800	5359369022865	3926646208485	1892225972637	578157321105	107603516475	11550870135	645700815	14348907		22073545867264
14	211799312	5943756288	75257843040	567466966080	2832331811400	9837576953568	24335417178072	43156331515440	54543345475905	48326177928600	29265340788684	11740921939656	3018218185710	478998402120	44481611700	2181033864	43046721	228230246236160

Technical details

Node id: n-3e2e1a8ae420fb53 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-419a54f90e96411a

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:      4,      15,        9;
  1:     10,      54,       81,        27;
  2:     26,     192,      459,       378,        81;
  3:     76,     690,     2250,      3105,      1620,       243;
  4:    232,    2538,    10530,     20385,     18225,      6561,       729;
  5:    764,    9660,    48384,    120960,    155925,     96957,     25515,      2187;
  6:   2620,   37824,   222516,    681912,   1153845,   1054620,    479682,     96228,      6561;
  7:   9496,  153252,  1032264,   3744468,   7868826,   9598743,   6491016,   2243862,    354294,    19683;
  8:  35696,  638580,  4851900,  20335860,  51098040,  78366771,  71722665,  37069650,  10038330,  1279395,   59049;
  9: 140152, 2744808, 23179860, 110145420, 322230150, 596918322, 694982673, 491123655, 199191960, 43302600, 4546773, 177147;

%F Formula

The row polynomials satisfy P(n,x) = (n+2+(3*n+6)*x)*P(n-2,x) + (1+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 4 + 15*x + 9*x^2, P(1,x) = 10 + 54*x + 81*x^2 + 27*x^3.

%t Mathematica

P[0, x_] := 4 + 15*x + 9*x^2;
P[1, x_] := 10 + 54*x + 81*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[(n + 2 + (3*n + 6)*x)*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] = 4 + 15*x + 9*x^2; v[2] = 10 + 54*x + 81*x^2 + 27*x^3; for(n = 2, N, my(i = n + 1); v[i] = (n + 2 + (3*n + 6)*x)*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 A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= 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

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