A146986 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(24 + 48\,x + 24\,x^{2} - 8\,n - 16\,n x - 8\,n x^{2}\right)\,P_{n-2}(x) + \left(-18 - 18\,x + 6\,n + 6\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 6 + 12\,x + 4\,x^{2}\)

\(P_{3}(x) = 16 + 48\,x + 40\,x^{2} + 8\,x^{3}\)

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	6	12	4													22
3	16	48	40	8												112
4	48	192	256	128	16											640
5	160	800	1472	1216	416	32										4096
6	576	3456	8128	9472	5568	1408	64									28672
7	2176	15232	43648	65920	55680	25216	4992	128								212992
8	8448	67584	228352	423936	468480	309248	113664	18432	256							1638400
9	33280	299520	1165312	2566144	3505152	3046400	1648640	509952	70144	512						12845056
10	132096	1320960	5813248	14802944	24070144	25948160	18565120	8511488	2274304	272384	1024					101711872
11	526336	5789696	28424192	82126848	154816512	199127040	177106944	107630592	42805248	10074112	1071104	2048				809500672
12	2101248	25214976	136585216	441303040	945745920	1412530176	1501151232	1135706112	599715840	210616320	44310528	4243456	4096			6459228160
13	8396800	109158400	646561792	2309210112	5542338560	9422561280	11640668160	10533371904	6931144704	3235471360	1017364480	193576960	16883712	8192		51606716416
14	33570816	469991424	3021389824	11817123840	31389794304	59826798592	84203716608	88639930368	69808865280	40633663488	16994942976	4837801984	840351744	67338240	16384	412585295872

Technical details

Node id: n-e34f9664814ccfa0 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-9e2859b846aec920

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:     6,     12,       4;
  3:    16,     48,      40,       8;
  4:    48,    192,     256,     128,      16;
  5:   160,    800,    1472,    1216,     416,      32;
  6:   576,   3456,    8128,    9472,    5568,    1408,      64;
  7:  2176,  15232,   43648,   65920,   55680,   25216,    4992,    128;
  8:  8448,  67584,  228352,  423936,  468480,  309248,  113664,  18432,   256;
  9: 33280, 299520, 1165312, 2566144, 3505152, 3046400, 1648640, 509952, 70144, 512;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = -(8*n-24+(16*n-48)*x+(8*n-24)*x^2)*P(n-2,x) + (6*n-18+(6*n-18)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 6 + 12*x + 4*x^2, P(3,x) = 16 + 48*x + 40*x^2 + 8*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 6 + 12*x + 4*x^2;
P[3, x_] := 16 + 48*x + 40*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 24 + (16*n - 48)*x + (8*n - 24)*x^2)*P[n - 2, x] + (6*n - 18 + (6*n - 18)*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] = 6 + 12*x + 4*x^2; v[4] = 16 + 48*x + 40*x^2 + 8*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(8*n - 24 + (16*n - 48)*x + (8*n - 24)*x^2)*v[i - 2] + (6*n - 18 + (6*n - 18)*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