A269951 → basis A013611

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	4														5
3	3	16	16													35
4	11	72	128	64												275
5	49	372	896	832	256											2405
6	261	2212	6480	8384	4864	1024										23225
7	1631	15040	50624	81280	66560	26624	4096									245855
8	11743	115784	432752	810816	833280	474112	139264	16384								2834135
9	95901	998788	4054160	8540672	10343424	7440384	3125248	704512	65536							35368625
10	876809	9560996	41507488	95947648	131763968	112458752	60039168	19431424	3473408	262144						475321805
11	8877691	100687888	462393072	1153128704	1750446848	1704579072	1085263872	448266240	115408896	16777216	1048576					6846878075
12	98641011	1157716456	5577901024	14823587904	24432001280	26444607488	19301130240	9545367552	3146711040	660602880	79691776	4194304				105272152955
13	1193556233	14437999092	72519648416	203486959808	359405899776	424462111744	345840484352	196811636736	77934624768	21006909440	3667918848	373293056	16777216			1721157819485
14	15624736141	194171331204	1011670162928	2975836564032	5576878422016	7089202481152	6327302795264	4033725808640	1843717472256	598568599552	134501892096	19851640832	1728053248	67108864		29822847068225
15	220048367319	2801812923520	15081501987744	46242385594496	91243651789056	123552005564416	119116266041344	83453358669824	42855449821184	16101315969024	4367787753472	831302664192	105126035456	7918845952	268435456	545980200462455

Technical details

Node id: n-66cdc186e4602e6f · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-f4880ddb0267db87

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:      1;
   2:      1,       4;
   3:      3,      16,       16;
   4:     11,      72,      128,       64;
   5:     49,     372,      896,      832,       256;
   6:    261,    2212,     6480,     8384,      4864,      1024;
   7:   1631,   15040,    50624,    81280,     66560,     26624,     4096;
   8:  11743,  115784,   432752,   810816,    833280,    474112,   139264,    16384;
   9:  95901,  998788,  4054160,  8540672,  10343424,   7440384,  3125248,   704512,   65536;
  10: 876809, 9560996, 41507488, 95947648, 131763968, 112458752, 60039168, 19431424, 3473408, 262144;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n - 3)*P[n - 2, x] + (n + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = -(n - 3)*v[i - 2] + (n + 4*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for 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