A008556 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-4 - 12\,x + 4\,n + 12\,n x\right)\,P_{n-2}(x) + \left(-2 + 4\,n\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	8	6							14
3	32	36							68
4	136	216	54						406
5	592	1200	540						2332
6	2624	6480	4320	540					13964
7	11776	34272	30240	7560					83848
8	53344	178752	196560	75600	5670				509926
9	243392	922752	1215648	635040	102060				3118892
10	1116928	4726080	7257600	4808160	1224720	61236			19194724
11	5149696	24055680	42197760	33929280	11975040	1347192			118654648
12	23835904	121828608	240356160	227525760	103284720	18860688	673596		736365436
13	110690816	614435328	1346716800	1467797760	817296480	210161952	17513496		4584612632
14	515483648	3088071168	7444749312	9184855680	6071345280	2031565536	280215936	7505784	28623792344

Technical details

Node id: n-60114611aeb15c63 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-f431603ae9ccf22a

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,      6;
  3:     32,     36;
  4:    136,    216,      54;
  5:    592,   1200,     540;
  6:   2624,   6480,    4320,    540;
  7:  11776,  34272,   30240,   7560;
  8:  53344, 178752,  196560,  75600,   5670;
  9: 243392, 922752, 1215648, 635040, 102060;

%F Formula

The row polynomials satisfy n*P(n,x) = (4*n-4+(12*n-12)*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 + (12*n - 12)*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 + (12*n - 12)*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