A153656 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-966 - 2898\,x + 483\,n + 1449\,n x\right)\,P_{n-2}(x) + \left(-4 - 6\,x + 2\,n + 3\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 46 + 69\,x\)

\(P_{2}(x) = 1058 + 3174\,x + 18\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	46	69														115
2	1058	3174	18													4250
3	24334	109503	109539	54												243430
4	559682	3358092	5155407	354807	162											9428150
5	12872686	96545145	231962274	174923928	1142991	486										517447510
6	296071778	2664646002	8109496872	8687291202	1041251355	3664683	1458									20802423350
7	6809650894	71501334387	276144625197	462304665270	282161212641	4787512128	11701179	4374								1103720706070
8	156621970562	1879463646744	8544752299629	17699665824225	15042045777255	2366601917463	19696768623	37224927	13122							45708885442550
9	3602305322926	48631121859501	259711183036404	684460303783200	889346414709018	461023306641324	14081956032564	76121876736	118038951	39366						2360932831339990
10	82853022427298	1242795336409470	7515773916504174	23078338831938708	36744197630483826	26529078914786826	4849953549637896	70956709225632	282546965259	373209363	118098					100114230831706550
11	1905619515827854	31442722011159591	214666930823236839	775619830360424934	1564060710243096270	1674627962742151704	764111499930643158	35133295225013676	323792833811211	1019444768280	1176905619	354294				5061893384307393430
12	43829248864040642	788926479552731556	5954591271808790523	24232434514031504475	56642941334570343192	74097325538626693050	47335269800088956070	9424455874223248386	209000213194824315	1383008203752813	3601525524399	3702549447	1062882			218730160888394021750
13	1008072723872934766	19657418115522227937	163520593764069416310	752005403670444803352	2065297643450906427585	3393292750227670183938	3112564288695936455700	1285021666141896906432	79755904768835025225	1099434864007502550	5633971576958934	12517489038024	11623146111	3188646		10873228822923854215510
14	23185672649077499618	486899125630627491978	4405235495030259363684	22327041211761251984886	68857949773342273011237	130847108659757179678989	146634966846987927474474	85048554325950329495832	17771560472685947825205	544975885916233121889	5315290959242536272	22147348525189902	42941220927003	36415931643	9565938	476952814950996521098550

Technical details

Node id: n-2f014dad76c8185f · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-b0048c17ed6c5e8a

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:            46,             69;
  2:          1058,           3174,              18;
  3:         24334,         109503,          109539,              54;
  4:        559682,        3358092,         5155407,          354807,             162;
  5:      12872686,       96545145,       231962274,       174923928,         1142991,             486;
  6:     296071778,     2664646002,      8109496872,      8687291202,      1041251355,         3664683,           1458;
  7:    6809650894,    71501334387,    276144625197,    462304665270,    282161212641,      4787512128,       11701179,        4374;
  8:  156621970562,  1879463646744,   8544752299629,  17699665824225,  15042045777255,   2366601917463,    19696768623,    37224927,     13122;
  9: 3602305322926, 48631121859501, 259711183036404, 684460303783200, 889346414709018, 461023306641324, 14081956032564, 76121876736, 118038951, 39366;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (483*n-966+(1449*n-2898)*x)*P(n-2,x) + (2*n-4+(3*n-6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 46 + 69*x, P(2,x) = 1058 + 3174*x + 18*x^2.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 46 + 69*x;
P[2, x_] := 1058 + 3174*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((483*n - 966 + (1449*n - 2898)*x)*P[n - 2, x] + (2*n - 4 + (3*n - 6)*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] = 46 + 69*x; v[3] = 1058 + 3174*x + 18*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((483*n - 966 + (1449*n - 2898)*x)*v[i - 2] + (2*n - 4 + (3*n - 6)*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) = T(n-1, k) + T(n-1, k-1) + (2*j +3)*prime(j)*T(n-2, k-1) with j=9, read by rows.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? 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