A153653 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-646 - 646\,x + 323\,n + 323\,n x\right)\,P_{n-2}(x) + \left(-4 - 2\,x + 2\,n + n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 38 + 19\,x\)

\(P_{2}(x) = 722 + 722\,x + 2\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	38	19														57
2	722	722	2													1446
3	13718	20577	6863	2												41160
4	260642	521284	268155	7513	2											1057596
5	4952198	12380495	9920714	2500576	8163	2										29762148
6	94091762	282275286	287210720	103962630	4944247	8813	2									772493460
7	1787743478	6257102173	8059987233	4507212650	924173821	7599168	9463	2								21543827988
8	33967126082	135868504328	206321056577	143423404583	41532481563	2539210537	10465339	10113	2							563662259124
9	645375395558	2904189280011	5172930495620	4552373427952	1980826197842	347573578084	5017729028	13542760	10763	2						15608299657620
10	12262132515602	61310662578010	123777278443566	127245138306204	68254777048794	16911129902310	1181154344088	8428385544	16831431	11413	2					410950718366964
11	232980517796438	1281392847880409	2917774906993955	3519540722369730	2373978171535254	854150164477512	133160430787662	2823111902700	12839836335	20331352	12063	2				11315813733923412
12	4426629838132322	26559779028793932	66900347611837367	91037096961909125	71413969725104592	31590866445696870	6964278837639390	523041876264198	5576596658295	18320737651	24042523	12713	2			299421605266828980
13	84105966924514118	546688785009341767	1516691659076888238	2328227489820125872	2137471639143428141	1177281155148621750	364420746391626852	51933046879154712	1450207481289093	9767072296498	24939745742	27964944	13363	2		8208280487915011092
14	1598013371565768242	11186093600960377694	33767693008087027820	57186941236037256898	59074865308012412745	37762596032609543867	14359434574449487434	2886691430740795200	225577228595698137	3276899944772647	15742315784184	32765516858	32098615	14013	2	218051198466116554356

Technical details

Node id: n-3a242ada054de5d6 · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-9b155509a8b144c4

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:           38,            19;
  2:          722,           722,             2;
  3:        13718,         20577,          6863,             2;
  4:       260642,        521284,        268155,          7513,             2;
  5:      4952198,      12380495,       9920714,       2500576,          8163,            2;
  6:     94091762,     282275286,     287210720,     103962630,       4944247,         8813,          2;
  7:   1787743478,    6257102173,    8059987233,    4507212650,     924173821,      7599168,       9463,        2;
  8:  33967126082,  135868504328,  206321056577,  143423404583,   41532481563,   2539210537,   10465339,    10113,     2;
  9: 645375395558, 2904189280011, 5172930495620, 4552373427952, 1980826197842, 347573578084, 5017729028, 13542760, 10763, 2;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (323*n-646+(323*n-646)*x)*P(n-2,x) + (2*n-4+(n-2)*x)*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 38 + 19*x, P(2,x) = 722 + 722*x + 2*x^2.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 38 + 19*x;
P[2, x_] := 722 + 722*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((323*n - 646 + (323*n - 646)*x)*P[n - 2, x] + (2*n - 4 + (n - 2)*x)*P[n - 1, x])/(n - 2)]];
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] = 38 + 19*x; v[3] = 722 + 722*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((323*n - 646 + (323*n - 646)*x)*v[i - 2] + (2*n - 4 + (n - 2)*x)*v[i - 1])/(n - 2)); 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and…?
  - 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