A153518 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(-15 - 60\,x + 5\,n + 20\,n x\right)\,P_{n-2}(x) + \left(-6 - 12\,x + 2\,n + 4\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 10 + 20\,x\)

\(P_{2}(x) = 50 + 200\,x + 32\,x^{2}\)

\(P_{3}(x) = 250 + 1500\,x + 2064\,x^{2} + 128\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	10	20														30
2	50	200	32													282
3	250	1500	2064	128												3942
4	750	6000	14288	9152	512											30702
5	2750	27500	92896	117376	40192	2048										282762
6	9250	111000	487232	937856	735488	175104	8192									2464122
7	32250	451500	2432944	6269440	7770880	4106240	757760	32768								21853782
8	110750	1772000	11328048	36704576	63054080	54881280	21483520	3260416	131072							192725742
9	382750	6889500	50938816	198727424	437169664	537927680	348405760	107773952	13959168	524288						1702699002
10	1319250	26385000	221515872	1011293952	2718610944	4360022016	4053565440	2055143424	524877824	59506688	2097152					15034337562
11	4552250	100149500	941055824	4921064832	15642794496	31027519488	38047801344	27831533568	11495604224	2500329472	252706816	8388608				132773500422
12	15700750	376818000	3917989008	23093140160	84788782080	200798545920	307673948160	299201298432	178044600320	61778165760	11707351040	1069547520	33554432			1172499441582
13	54162750	1408231500	16051519136	105284676992	438585393920	1208745707520	2229331476480	2729212084224	2167003086848	1078148464640	321797488640	54064578560	4513071104	134217728		10354334160042
14	186829250	5231219000	64862269312	468600911488	2184116209408	6871601361920	14847986442240	22025235529728	22125103480832	14694092111872	6250288906240	1634813870080	246843179008	18991808512	536870912	91438490999802

Technical details

Node id: n-98886efaea10cd66 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-2aa284a691d23675

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:      2;
  1:     10,      20;
  2:     50,     200,       32;
  3:    250,    1500,     2064,       128;
  4:    750,    6000,    14288,      9152,       512;
  5:   2750,   27500,    92896,    117376,     40192,      2048;
  6:   9250,  111000,   487232,    937856,    735488,    175104,      8192;
  7:  32250,  451500,  2432944,   6269440,   7770880,   4106240,    757760,     32768;
  8: 110750, 1772000, 11328048,  36704576,  63054080,  54881280,  21483520,   3260416,   131072;
  9: 382750, 6889500, 50938816, 198727424, 437169664, 537927680, 348405760, 107773952, 13959168, 524288;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (5*n-15+(20*n-60)*x)*P(n-2,x) + (2*n-6+(4*n-12)*x)*P(n-1,x) for n >= 4, with P(0,x) = 2, P(1,x) = 10 + 20*x, P(2,x) = 50 + 200*x + 32*x^2, P(3,x) = 250 + 1500*x + 2064*x^2 + 128*x^3.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 10 + 20*x;
P[2, x_] := 50 + 200*x + 32*x^2;
P[3, x_] := 250 + 1500*x + 2064*x^2 + 128*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((5*n - 15 + (20*n - 60)*x)*P[n - 2, x] + (2*n - 6 + (4*n - 12)*x)*P[n - 1, x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 10 + 20*x; v[3] = 50 + 200*x + 32*x^2; v[4] = 250 + 1500*x + 2064*x^2 + 128*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((5*n - 15 + (20*n - 60)*x)*v[i - 2] + (2*n - 6 + (4*n - 12)*x)*v[i - 1])/(n - 3)); 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 ≥ 3
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.?
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - 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