A146988 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(48 + 96\,x + 48\,x^{2} - 16\,n - 32\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(-30 - 30\,x + 10\,n + 10\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 8 + 16\,x + 4\,x^{2}\)

\(P_{3}(x) = 40 + 120\,x + 88\,x^{2} + 8\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	2							4
2	8	16	4						28
3	40	120	88	8					256
4	272	1088	1376	576	16				3328
5	2080	10400	18752	14656	4256	32			50176
6	16448	98688	230336	263424	148416	33152	64		790528
7	131200	918400	2624128	3936640	3281280	1444480	263040	128	12599296

All 15 rows

Technical details

Node id: n-311227396374f6e7 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-58471b9a07a8bbc9

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,        2;
  2:       8,       16,         4;
  3:      40,      120,        88,         8;
  4:     272,     1088,      1376,       576,        16;
  5:    2080,    10400,     18752,     14656,      4256,        32;
  6:   16448,    98688,    230336,    263424,    148416,     33152,        64;
  7:  131200,   918400,   2624128,   3936640,   3281280,   1444480,    263040,       128;
  8: 1048832,  8390656,  28318720,  52443136,  57689600,  37763072,  13638656,   2099200,      256;
  9: 8389120, 75502080, 293619712, 645965824, 880868352, 763427840, 411084800, 125847552, 16781824, 512;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = -(16*n-48+(32*n-96)*x+(16*n-48)*x^2)*P(n-2,x) + (10*n-30+(10*n-30)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 8 + 16*x + 4*x^2, P(3,x) = 40 + 120*x + 88*x^2 + 8*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 8 + 16*x + 4*x^2;
P[3, x_] := 40 + 120*x + 88*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n - 48 + (32*n - 96)*x + (16*n - 48)*x^2)*P[n - 2, x] + (10*n - 30 + (10*n - 30)*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 + 2*x; v[3] = 8 + 16*x + 4*x^2; v[4] = 40 + 120*x + 88*x^2 + 8*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(16*n - 48 + (32*n - 96)*x + (16*n - 48)*x^2)*v[i - 2] + (10*n - 30 + (10*n - 30)*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) + 4^(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