A115951 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(4 + 2\,x - 2\,n\right)\,P_{n-1}(x) + \left(8 + 12\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

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	2														4
2	8	14	6													28
3	32	84	72	20												208
4	136	472	606	340	70											1624
5	592	2560	4380	3700	1540	252										13024
6	2624	13584	29040	32780	20580	6804	924									106336
7	11776	71008	182112	257320	216160	107856	29568	3432								879232
8	53344	367168	1098384	1864240	1962310	1310904	542388	126984	12870							7338592
9	243392	1882944	6436512	12754560	16139340	13517532	7489944	2646072	540540	48620						61699456
10	1116928	9593920	36889920	83591520	123572400	124479684	86500260	40926600	12612600	2285140	184756					521753728
11	5149696	48628096	207736320	529805760	896058240	1054951128	881950608	523397160	216010080	59024680	9607312	705432				4433024512
12	23835904	245421312	1153171008	3269009920	6225413040	8388393552	8198383116	5854398264	3030704820	1108924960	272145588	40209624	2704156			37812715264
13	110690816	1234168832	6325777536	19731620480	41781740000	63410675328	70946932056	59238135528	36903592440	16937165960	5565220232	1239444024	167657672	10400600		323603221504
14	515483648	6187414016	34354564608	116932350080	272530818560	460022026464	579844513248	554338448664	403834190760	223047264440	91924608776	27406738632	5586786296	696840200	40116600	2777262164992

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

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

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

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

Technical details

Node id: n-234411cdfee41c66 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-f8f6fd71656d994a)
Fingerprint: f6-38273efd8b18cf00

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,       2;
  2:      8,      14,       6;
  3:     32,      84,      72,       20;
  4:    136,     472,     606,      340,       70;
  5:    592,    2560,    4380,     3700,     1540,      252;
  6:   2624,   13584,   29040,    32780,    20580,     6804,     924;
  7:  11776,   71008,  182112,   257320,   216160,   107856,   29568,    3432;
  8:  53344,  367168, 1098384,  1864240,  1962310,  1310904,  542388,  126984,  12870;
  9: 243392, 1882944, 6436512, 12754560, 16139340, 13517532, 7489944, 2646072, 540540, 48620;

%F Formula

The row polynomials satisfy n*P(n,x) = -(2*n-4-2*x)*P(n-1,x) + (8+12*x+4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 - 2*x)*P[n - 1, x] + (8 + 12*x + 4*x^2)*D[P[n - 1, x], 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; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n - 4 - 2*x)*v[i - 1] + (8 + 12*x + 4*x^2)*deriv(v[i - 1], x))/(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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by 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
Ore-algebra identity of two definitions 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 (above)

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