A008556 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-4 - 16\,x + 4\,n + 16\,n x\right)\,P_{n-2}(x) + \left(-2 + 4\,n\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	8	8							16
3	32	48							80
4	136	288	96						520
5	592	1600	960						3152
6	2624	8640	7680	1280					20224
7	11776	45696	53760	17920					129152
8	53344	238336	349440	179200	17920				838240
9	243392	1230336	2161152	1505280	322560				5462720
10	1116928	6301440	12902400	11397120	3870720	258048			35846656
11	5149696	32074240	75018240	80424960	37847040	5677056			236191232
12	23835904	162438144	427299840	539320320	326430720	79478784	3784704		1562588416
13	110690816	819247104	2394163200	3479224320	2583060480	885620736	98402304		10370408960
14	515483648	4117428224	13235109888	21771509760	19188449280	8561000448	1574436864	56229888	69019648000

Technical details

Node id: n-6f557a1abb6b17db · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-2ae7124585c92266

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:      8,       8;
  3:     32,      48;
  4:    136,     288,      96;
  5:    592,    1600,     960;
  6:   2624,    8640,    7680,    1280;
  7:  11776,   45696,   53760,   17920;
  8:  53344,  238336,  349440,  179200,  17920;
  9: 243392, 1230336, 2161152, 1505280, 322560;

%F Formula

The row polynomials satisfy n*P(n,x) = (4*n-4+(16*n-16)*x)*P(n-2,x) + (4*n-2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (16*n - 16)*x)*P[n - 2, x] + (4*n - 2)*P[n - 1, x])/(n)]];
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; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (16*n - 16)*x)*v[i - 2] + (4*n - 2)*v[i - 1])/(n)); 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 Triangle of coefficients of Legendre polynomials 2^n P_n (x).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Expansion of 1/sqrt(1-4*x*y-4*x^2*y).?
    
    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