A228955 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4\,n x + 4\,n^{2} x\right)\,P_{n-2}(x) + \left(2\,n\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	4							6
2	8	24							32
3	48	192	96						336
4	384	1920	1920						4224
5	3840	23040	34560	7680					69120
6	46080	322560	645120	322560					1336320
7	645120	5160960	12902400	10321920	1290240				30320640
8	10321920	92897280	278691840	309657600	92897280				784465920
9	185794560	1857945600	6502809600	9289728000	4644864000	371589120			22852730880
10	3715891200	40874803200	163499212800	286123622400	204374016000	40874803200			739462348800
11	81749606400	980995276800	4414478745600	9155955916800	8583708672000	2942985830400	163499212800		26323373260800
12	1961990553600	25505877196800	127529385984000	306070526361600	357082280755200	178541140377600	25505877196800		1022197078425600
13	51011754393600	714164561510400	3927905088307200	10712468422656000	14997455791718400	9998303861145600	2499575965286400	102023508787200	43002908953804800
14	1428329123020800	21424936845312000	128549621071872000	392790508830720000	642748105359360000	539908408501862400	199966077222912000	21424936845312000	1948240923800371200

Technical details

Node id: n-cf77e3825db15ef0 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-088313a0be276231

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,          4;
  2:         8,         24;
  3:        48,        192,         96;
  4:       384,       1920,       1920;
  5:      3840,      23040,      34560,       7680;
  6:     46080,     322560,     645120,     322560;
  7:    645120,    5160960,   12902400,   10321920,    1290240;
  8:  10321920,   92897280,  278691840,  309657600,   92897280;
  9: 185794560, 1857945600, 6502809600, 9289728000, 4644864000, 371589120;

%F Formula

The row polynomials satisfy P(n,x) = (4*n^2-4*n)*x*P(n-2,x) + 2*n*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[(4*n^2 - 4*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (4*n^2 - 4*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