A228955 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3\,n x + 3\,n^{2} x\right)\,P_{n-2}(x) + \left(2\,n\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	3							5
2	8	18							26
3	48	144	54						246
4	384	1440	1080						2904
5	3840	17280	19440	3240					43800
6	46080	241920	362880	136080					786960
7	645120	3870720	7257600	4354560	408240				16536240
8	10321920	69672960	156764160	130636800	29393280				396789120
9	185794560	1393459200	3657830400	3919104000	1469664000	88179840			10714032000
10	3715891200	30656102400	91968307200	120708403200	64665216000	9699782400			321413702400
11	81749606400	735746457600	2483144294400	3862668902400	2715939072000	698384332800	29099347200		10606732012800
12	1961990553600	19129407897600	71735279616000	129123503308800	112983065395200	42368649523200	4539498163200		381841394457600
13	51011754393600	535623421132800	2209446612172800	4519322615808000	4745288746598400	2372644373299200	444870819993600	13618494489600	14891826837888000
14	1428329123020800	16068702633984000	72309161852928000	165708495912960000	203369517711360000	128122796158156800	35589665599488000	2859883842816000	625456552834713600

Technical details

Node id: n-ced9d9392f88079d · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-f03a94f3319f9845

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half affine: slope 1, intercept 1)
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,          3;
  2:         8,         18;
  3:        48,        144,         54;
  4:       384,       1440,       1080;
  5:      3840,      17280,      19440,       3240;
  6:     46080,     241920,     362880,     136080;
  7:    645120,    3870720,    7257600,    4354560,     408240;
  8:  10321920,   69672960,  156764160,  130636800,   29393280;
  9: 185794560, 1393459200, 3657830400, 3919104000, 1469664000, 88179840;

%F Formula

The row polynomials satisfy P(n,x) = (3*n^2-3*n)*x*P(n-2,x) + 2*n*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[(3*n^2 - 3*n)*x*P[n - 2, x] + 2*n*P[n - 1, x]];
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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (3*n^2 - 3*n)*x*v[i - 2] + 2*n*v[i - 1]); 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 Table: T(n,k) = n!*binomial(n+1,2*k).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

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