A153651 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-286 - 572\,x + 143\,n + 286\,n x\right)\,P_{n-2}(x) + \left(-4 - 4\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 338 + 676\,x + 8\,x^{2}\)

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	26														52
2	338	676	8													1022
3	4394	13182	8804	16												26396
4	57122	228488	238452	19928	32											544022
5	742586	3712930	5962904	3036992	44496	64										13499972
6	9653618	57921708	118797872	89046768	11866960	98272	128									287385326
7	125497034	878479238	2268032412	2555369680	1076770096	36665472	215104	256								6941029292
8	1631461442	13051691536	39846727484	56356683400	34428630480	5634874592	101885248	467328	512							151052422022
9	21208998746	190880988714	681370535024	1206481955840	1066384479968	393326420096	21990604544	266261504	1008896	1024						3581911254356
10	275716983698	2757169836980	11175368856984	23230874768352	25587038482656	13571797184064	2456777773056	73719740928	668269824	2166272	2048					79129134064862
11	3584320788074	39427528668814	179893026668564	436211379952656	595182646507680	439549310678016	147693162511488	11388383322624	225071083776	1629562880	4629504	4096				1853156464378172
12	46596170244962	559154042939544	2825269430599748	7750379398214200	12365764739689152	11328123917732160	5407338162568320	1031343457714944	44406317303040	644836239872	3888231424	9852928	8192			41359024371338486
13	605750213184506	7874752771398578	43769822959964040	134978930618067008	250099861392864560	280465565642995392	180302149253656320	56746146533987328	5440762345286400	154705664537600	1764165945344	9120792576	20893696	16384		960440220703589732
14	7874752771398578	110246538799580092	667220736319198784	2273828818252220624	4755070449686673072	6217661289858516256	4934628227511970944	2168077420523063808	425688150039356160	23983354350619136	497918842662912	4660016631808	21102481408	44163072	32768	21584782338118569422

Technical details

Node id: n-85cb74c94a93cf11 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-c52a40314e6c14a2

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,           26;
  2:         338,          676,            8;
  3:        4394,        13182,         8804,            16;
  4:       57122,       228488,       238452,         19928,            32;
  5:      742586,      3712930,      5962904,       3036992,         44496,           64;
  6:     9653618,     57921708,    118797872,      89046768,      11866960,        98272,         128;
  7:   125497034,    878479238,   2268032412,    2555369680,    1076770096,     36665472,      215104,       256;
  8:  1631461442,  13051691536,  39846727484,   56356683400,   34428630480,   5634874592,   101885248,    467328,     512;
  9: 21208998746, 190880988714, 681370535024, 1206481955840, 1066384479968, 393326420096, 21990604544, 266261504, 1008896, 1024;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (143*n-286+(286*n-572)*x)*P(n-2,x) + (2*n-4+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 26 + 26*x, P(2,x) = 338 + 676*x + 8*x^2.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 26 + 26*x;
P[2, x_] := 338 + 676*x + 8*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((143*n - 286 + (286*n - 572)*x)*P[n - 2, x] + (2*n - 4 + (2*n - 4)*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 + 26*x; v[3] = 338 + 676*x + 8*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((143*n - 286 + (286*n - 572)*x)*v[i - 2] + (2*n - 4 + (2*n - 4)*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