A146986 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(24 + 96\,x + 96\,x^{2} - 8\,n - 32\,n x - 32\,n x^{2}\right)\,P_{n-2}(x) + \left(-18 - 36\,x + 6\,n + 12\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 4\,x\)

\(P_{2}(x) = 6 + 24\,x + 16\,x^{2}\)

\(P_{3}(x) = 16 + 96\,x + 160\,x^{2} + 64\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	6	24	16													46
3	16	96	160	64												336
4	48	384	1024	1024	256											2736
5	160	1600	5888	9728	6656	1024										25056
6	576	6912	32512	75776	89088	45056	4096									254016
7	2176	30464	174592	527360	890880	806912	319488	16384								2768256
8	8448	135168	913408	3391488	7495680	9895936	7274496	2359296	65536							31539456
9	33280	599040	4661248	20529152	56082432	97484800	105512960	65273856	17956864	262144						368395776
10	132096	2641920	23252992	118423552	385122304	830341120	1188167680	1089470464	582221824	139460608	1048576					4360283136
11	526336	11579392	113696768	657014784	2477064192	6372065280	11334844416	13776715776	10958143488	5157945344	1096810496	4194304				51960600576
12	2101248	50429952	546340864	3530424320	15131934720	45200965632	96073678848	145370382336	153527255040	107835555840	45373980672	8690597888	16777216			621350424576
13	8396800	218316800	2586247168	18473680896	88677416960	301521960960	745002762240	1348271603712	1774373044224	1656561336320	1041781227520	396445614080	69155684352	67108864		7443144400896
14	33570816	939982848	12085559296	94536990720	502236708864	1914457554944	5389037862912	11345911087104	17871069511680	20804435705856	17402821607424	9907818463232	3442080743424	551634862080	268435456	89239368646656

Technical details

Node id: n-29934602e3835baa · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-a6c102f752e8137e

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,      4;
  2:     6,     24,      16;
  3:    16,     96,     160,       64;
  4:    48,    384,    1024,     1024,      256;
  5:   160,   1600,    5888,     9728,     6656,     1024;
  6:   576,   6912,   32512,    75776,    89088,    45056,      4096;
  7:  2176,  30464,  174592,   527360,   890880,   806912,    319488,    16384;
  8:  8448, 135168,  913408,  3391488,  7495680,  9895936,   7274496,  2359296,    65536;
  9: 33280, 599040, 4661248, 20529152, 56082432, 97484800, 105512960, 65273856, 17956864, 262144;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = -(8*n-24+(32*n-96)*x+(32*n-96)*x^2)*P(n-2,x) + (6*n-18+(12*n-36)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 6 + 24*x + 16*x^2, P(3,x) = 16 + 96*x + 160*x^2 + 64*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 6 + 24*x + 16*x^2;
P[3, x_] := 16 + 96*x + 160*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 24 + (32*n - 96)*x + (32*n - 96)*x^2)*P[n - 2, x] + (6*n - 18 + (12*n - 36)*x)*P[n - 1, x])/(n - 3)]];
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 + 4*x; v[3] = 6 + 24*x + 16*x^2; v[4] = 16 + 96*x + 160*x^2 + 64*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(8*n - 24 + (32*n - 96)*x + (32*n - 96)*x^2)*v[i - 2] + (6*n - 18 + (12*n - 36)*x)*v[i - 1])/(n - 3)); 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 ≥ 3
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.?
  - 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