A115951 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-4 - 12\,x + 4\,n + 12\,n x\right)\,P_{n-2}(x) + \left(-2 - 6\,x + 4\,n + 12\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 2 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	6							8
2	8	42	54						104
3	32	252	648	540					1472
4	136	1416	5454	9180	5670				21856
5	592	7680	39420	99900	124740	61236			333568
6	2624	40752	261360	885060	1666980	1653372	673596		5183744
7	11776	213024	1639008	6947640	17508960	26209008	21555072	7505784	81590272

All 15 rows

Technical details

Node id: n-2a67a442e762ff53 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-23e3cf183ada7515

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:      1;
  1:      2,       6;
  2:      8,      42,       54;
  3:     32,     252,      648,       540;
  4:    136,    1416,     5454,      9180,       5670;
  5:    592,    7680,    39420,     99900,     124740,      61236;
  6:   2624,   40752,   261360,    885060,    1666980,    1653372,     673596;
  7:  11776,  213024,  1639008,   6947640,   17508960,   26209008,   21555072,    7505784;
  8:  53344, 1101504,  9885456,  50334480,  158947110,  318549672,  395400852,  277714008,   84440070;
  9: 243392, 5648832, 57928608, 344373120, 1307286540, 3284760276, 5460169176, 5786959464, 3546482940, 956987460;

%F Formula

The row polynomials satisfy n*P(n,x) = (4*n-4+(12*n-12)*x)*P(n-2,x) + (4*n-2+(12*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (12*n - 12)*x)*P[n - 2, x] + (4*n - 2 + (12*n - 6)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (12*n - 12)*x)*v[i - 2] + (4*n - 2 + (12*n - 6)*x)*v[i - 1])/(n)); 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 ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Expansion of 1/sqrt(1-4*x*y-4*x^2*y).?
  - 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