A153651 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-286 - 286\,x + 143\,n + 143\,n x\right)\,P_{n-2}(x) + \left(-4 - 2\,x + 2\,n + n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 26 + 13\,x\)

\(P_{2}(x) = 338 + 338\,x + 2\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	26	13														39
2	338	338	2													678
3	4394	6591	2201	2												13188
4	57122	114244	59613	2491	2											233472
5	742586	1856465	1490726	379624	2781	2										4472184
6	9653618	28960854	29699468	11130846	741685	3071	2									80189544
7	125497034	439239619	567008103	319421210	67298131	1145796	3361	2								1519613256
8	1631461442	6525845768	9961681871	7044585425	2151789405	176089831	1591957	3651	2							27493049352
9	21208998746	95440494357	170342633756	150810244480	66649029998	12291450628	343603196	2080168	3941	2						517088539272
10	275716983698	1378584918490	2793842214246	2903859346044	1599189905166	424118662002	38387152704	575935476	2610429	4231	2					9414277732488
11	3584320788074	19713764334407	44973256667141	54526422494082	37198915406730	13735915958688	2307705664242	88971744708	879183921	3182740	4521	2				176130155429256
12	46596170244962	279577021469772	706317357649937	968797424776775	772860296230572	354003872429130	84489658790130	8057370763398	173462176965	1259445781	3797101	4811	2			3220873897779336
13	605750213184506	3937376385699289	10942455739991010	16872366327258376	15631241337054035	8764548926343606	2817221082088380	443329269796776	21252977911275	302159501050	1722818306	4453512	5101	2		60035846146105224
14	7874752771398578	55123269399790046	166805184079799696	284228602281527578	297191903105417067	194301915308078633	77103566054874546	16938104847836436	1662844336091235	46842488966053	486248869788	2275398746	5151973	5391	2	1101277473203205768

Technical details

Node id: n-4caadb66e6b8750b · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-a24db77fa0fde1b5

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:          26,          13;
  2:         338,         338,            2;
  3:        4394,        6591,         2201,            2;
  4:       57122,      114244,        59613,         2491,           2;
  5:      742586,     1856465,      1490726,       379624,        2781,           2;
  6:     9653618,    28960854,     29699468,     11130846,      741685,        3071,         2;
  7:   125497034,   439239619,    567008103,    319421210,    67298131,     1145796,      3361,       2;
  8:  1631461442,  6525845768,   9961681871,   7044585425,  2151789405,   176089831,   1591957,    3651,    2;
  9: 21208998746, 95440494357, 170342633756, 150810244480, 66649029998, 12291450628, 343603196, 2080168, 3941, 2;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (143*n-286+(143*n-286)*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) = 26 + 13*x, P(2,x) = 338 + 338*x + 2*x^2.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 26 + 13*x;
P[2, x_] := 338 + 338*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((143*n - 286 + (143*n - 286)*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] = 26 + 13*x; v[3] = 338 + 338*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((143*n - 286 + (143*n - 286)*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) = T(n-1, k) + T(n-1, k-1) + (j+5)*prime(j)*T(n-2, k-1) with j=6, 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