A110552 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(4 + 3\,x - n - n x\right)\,P_{n-1}(x) + \left(2 + 3\,x + x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	10	13	4													27
3	36	68	41	8												153
4	136	332	294	113	16											891
5	528	1568	1816	1032	289	32										5265
6	2080	7248	10320	7720	3210	705	64									31347
7	8256	32960	55536	51360	28220	9228	1665	128								187353
8	32896	147904	287392	315952	215320	93268	25102	3841	256							1121931
9	131328	656384	1443584	1836800	1492064	803264	286832	65552	8705	512						6725025
10	524800	2885888	7082496	10228992	9637824	6195168	2753184	835728	165906	19457	1024					40330467
11	2098176	12588032	34090240	55065600	58995840	44048256	23399712	8848320	2334900	409620	43009	2048				241923753
12	8390656	54537216	161508864	288400640	346072320	294155136	181680576	82187424	27034920	6308060	991254	94209	4096			1451365371
13	33558528	234905600	755042304	1476507648	1960963840	1868663808	1314973440	692085504	272506608	79300320	16580872	2359320	204801	8192		8707660785
14	134225920	1006686208	3489820672	7415822336	10796504576	11396253440	8997001728	5398179072	2474156256	862203056	224921840	42598712	5537818	442369	16384	52244370387

Other recurrences

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

\(P_{n}(x) = \left(-8 - 8\,x - 2\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 3\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 2\,x\)

Technical details

Node id: n-1ba7f2cb937015e6 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-ee6cba041ad56d92)
Fingerprint: f6-02204b3e41d44970

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      1;
  1:      3,      2;
  2:     10,     13,       4;
  3:     36,     68,      41,       8;
  4:    136,    332,     294,     113,      16;
  5:    528,   1568,    1816,    1032,     289,     32;
  6:   2080,   7248,   10320,    7720,    3210,    705,     64;
  7:   8256,  32960,   55536,   51360,   28220,   9228,   1665,   128;
  8:  32896, 147904,  287392,  315952,  215320,  93268,  25102,  3841,  256;
  9: 131328, 656384, 1443584, 1836800, 1492064, 803264, 286832, 65552, 8705, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(n-4+(n-3)*x)*P(n-1,x) + (2+3*x+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[-(n - 4 + (n - 3)*x)*P[n - 1, x] + (2 + 3*x + x^2)*D[P[n - 1, x], x]];
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] = -(n - 4 + (n - 3)*x)*v[i - 1] + (2 + 3*x + x^2)*deriv(v[i - 1], x)); 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 A triangular array related to A077028 and distributing the values of A007582.?
  - 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 A triangular array related to A077028 and distributing the values of A007582.?
  - ↑ 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