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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	1048832	8390656	28318720	52443136	57689600	37763072	13638656	2099200	256							201392128
9	8389120	75502080	293619712	645965824	880868352	763427840	411084800	125847552	16781824	512						3221487616
10	67109888	671098880	2952836096	7516315648	12214028288	13153595392	9395456000	4295090176	1140896768	134227968	1024					51540656128
11	536872960	5905602560	28991141888	83752200192	157840723968	202938150912	180389572608	109522341888	43486881792	10200659968	1073764352	2048				824637915136
12	4294971392	51539656704	279173144576	901944033280	1932737310720	2886221266944	3066610434048	2319285583872	1224067706880	429497630720	90194583552	8589983744	4096			13194156310528
13	34359746560	446676705280	2645700493312	9448930394112	22677433180160	38551636992000	47622611435520	43087125970944	28346794696704	13228505128960	4157530685440	790274621440	68719583232	8192		211106299641856
14	274877923328	3848290926592	24739013115904	96757029208064	257010859393024	489832462974976	689393839816704	725677730562048	571471217737728	332602300203008	139088237314048	39582424563712	6871949164544	549756043264	16384	3377699988963328

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