A153651 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-286 - 858\,x + 143\,n + 429\,n x\right)\,P_{n-2}(x) + \left(-4 - 6\,x + 2\,n + 3\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 26 + 39\,x\)

\(P_{2}(x) = 338 + 1014\,x + 18\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	26	39														65
2	338	1014	18													1370
3	4394	19773	19809	54												44030
4	57122	342732	536517	67257	162											1003790
5	742586	5569395	13416534	10249848	225261	486										30204110
6	9653618	86882562	267295212	300532842	60076485	746253	1458									725188430
7	125497034	1317718857	5103072927	8624372670	5451148611	278428428	2450169	4374								20902693070
8	1631461442	19577537304	89655136839	190203806475	174294941805	42789828933	1160536653	7984737	13122							519321247310
9	21208998746	286321483071	1533083703804	4071876600960	5398571429838	2986822502604	250486729884	4549327416	25856901	39366						14552946672590
10	275716983698	4135754755470	25144579928214	78404202343188	129534382318446	103060834866486	27984234321216	1259570886012	17127024669	83278773	118098					369816486824270
11	3584320788074	59141293003221	404759310004269	1472213407340214	3013112147945130	3337827577961184	1682317429232418	194581205676396	5768325705681	62645871420	266960529	354294				10173367930842830
12	46596170244962	838731064409316	6356856218849433	26157530468972925	62601683994676332	86022941000278590	61592961258004770	17621469859551426	1138085343067365	24789671307423	224215016949	852254217	1062882			262401870117696590
13	605750213184506	11812129157097867	98482101659919090	455553890835976152	1266130548301376835	2129785389101496258	2053754168842429020	969561113045549112	139440788075875275	5947405459167150	101730698150994	788926290264	2710880541	3188646		7131175807030581710
14	7874752771398578	165369808199370138	1501246656718197264	7674172261601244606	24072544151538782427	47215365419863107819	56208499654003544034	37043635302218285532	10909921689094592835	922000710318821199	28712509512111612	403080061657662	2737969683093	8594995293	9565938	185749748742475358030

Technical details

Node id: n-6aab2c08337784b8 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-fc9cd9a0d3c1ee91

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,           39;
  2:         338,         1014,            18;
  3:        4394,        19773,         19809,            54;
  4:       57122,       342732,        536517,         67257,           162;
  5:      742586,      5569395,      13416534,      10249848,        225261,           486;
  6:     9653618,     86882562,     267295212,     300532842,      60076485,        746253,         1458;
  7:   125497034,   1317718857,    5103072927,    8624372670,    5451148611,     278428428,      2450169,       4374;
  8:  1631461442,  19577537304,   89655136839,  190203806475,  174294941805,   42789828933,   1160536653,    7984737,    13122;
  9: 21208998746, 286321483071, 1533083703804, 4071876600960, 5398571429838, 2986822502604, 250486729884, 4549327416, 25856901, 39366;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (143*n-286+(429*n-858)*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) = 26 + 39*x, P(2,x) = 338 + 1014*x + 18*x^2.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 26 + 39*x;
P[2, x_] := 338 + 1014*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((143*n - 286 + (429*n - 858)*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] = 26 + 39*x; v[3] = 338 + 1014*x + 18*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((143*n - 286 + (429*n - 858)*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) + (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