A115951 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-4 - 8\,x + 4\,n + 8\,n x\right)\,P_{n-2}(x) + \left(-2 - 4\,x + 4\,n + 8\,n x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	8	28	24													60
3	32	168	288	160												648
4	136	944	2424	2720	1120											7344
5	592	5120	17520	29600	24640	8064										85536
6	2624	27168	116160	262240	329280	217728	59136									1014336
7	11776	142016	728448	2058560	3458560	3451392	1892352	439296								12182400
8	53344	734336	4393536	14913920	31396960	41948928	34712832	16253952	3294720							147702528
9	243392	3765888	25746048	102036480	258229440	432561024	479356416	338697216	138378240	24893440						1803907584
10	1116928	19187840	147559680	668732160	1977158400	3983349888	5536016640	5238604800	3228825600	1169991680	189190144					22159733760
11	5149696	97256192	830945280	4238446080	14336931840	33758436096	56444838912	66994836480	55298580480	30220636160	9837887488	1444724736				273508669440
12	23835904	490842624	4612684032	26152079360	99606608640	268428593664	524696519424	749362977792	775860433920	567769579520	278677082112	82349309952	11076222976			3389106769920
13	110690816	2468337664	25303110144	157852963840	668507840000	2029141610496	4540603651584	7582481347584	9447319664640	8671828971520	5698785517568	2538381361152	686725824512	85201715200		42134712606720
14	515483648	12374828032	137418258432	935458800640	4360493096960	14720704846848	37110048847872	70955321428992	103381552834560	114200199393280	94130799386624	56129000718336	22883476668416	5708514918400	657270374400	525323149885440

Technical details

Node id: n-406d5bfd4a642364 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-32632a63e62b0230

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,       4;
  2:      8,      28,       24;
  3:     32,     168,      288,       160;
  4:    136,     944,     2424,      2720,      1120;
  5:    592,    5120,    17520,     29600,     24640,      8064;
  6:   2624,   27168,   116160,    262240,    329280,    217728,     59136;
  7:  11776,  142016,   728448,   2058560,   3458560,   3451392,   1892352,    439296;
  8:  53344,  734336,  4393536,  14913920,  31396960,  41948928,  34712832,  16253952,   3294720;
  9: 243392, 3765888, 25746048, 102036480, 258229440, 432561024, 479356416, 338697216, 138378240, 24893440;

%F Formula

The row polynomials satisfy n*P(n,x) = (4*n-4+(8*n-8)*x)*P(n-2,x) + (4*n-2+(8*n-4)*x)*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[Together[((4*n - 4 + (8*n - 8)*x)*P[n - 2, x] + (4*n - 2 + (8*n - 4)*x)*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (8*n - 8)*x)*v[i - 2] + (4*n - 2 + (8*n - 4)*x)*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 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