A110552 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-8 - 10\,x - 2\,x^{2}\right)\,P_{n-2}(x) + \left(-10\,x - 4\,x^{2}\right)\,P_{n-2}'(x) + \left(-2\,x^{2}\right)\,P_{n-2}''(x) + \left(6 + 3\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	10	17	4													31
3	36	117	65	8												226
4	136	755	745	209	16											1861
5	528	4737	7415	3566	609	32										16887
6	2080	29267	68509	50365	14420	1665	64									166370
7	8256	179097	605255	639758	273525	52123	4353	128								1762495
8	32896	1089035	5194165	7602189	4559436	1276338	173859	11009	256							19939183
9	131328	6593697	43705055	86374646	69897429	26982186	5333706	545836	27137	512						239591532
10	524800	39805667	362716189	950952365	1011361780	517487355	139292538	20491350	1634358	65537	1024					3044332963
11	2098176	239825577	2980875095	10233823358	14037101925	9266983121	3263508171	647408256	73668210	4710465	155649	2048				40750160051
12	8390656	1442974715	24324730885	108293854669	188904872156	157806879516	70748356393	18180027723	2768143092	250964285	13156429	364545	4096			572742719160
13	33558528	8674102257	197475937295	1131555441926	2483392060149	2587062443680	1447568448612	468732436172	91722203859	11057436819	817679577	35790938	843777	8192		8428128391781
14	134225920	52110144467	1597123738669	11710335601965	32063458348740	41185298919765	28335024349632	11332258754960	2771897039626	426706604324	41739080383	2565919083	95207528	1933313	16384	129518749884759

Technical details

Node id: n-4840e5067f778103 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-e0a327ccba9473aa

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:      3,       2;
  2:     10,      17,        4;
  3:     36,     117,       65,        8;
  4:    136,     755,      745,      209,       16;
  5:    528,    4737,     7415,     3566,      609,       32;
  6:   2080,   29267,    68509,    50365,    14420,     1665,      64;
  7:   8256,  179097,   605255,   639758,   273525,    52123,    4353,    128;
  8:  32896, 1089035,  5194165,  7602189,  4559436,  1276338,  173859,  11009,   256;
  9: 131328, 6593697, 43705055, 86374646, 69897429, 26982186, 5333706, 545836, 27137, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(8+10*x+2*x^2)*P(n-2,x) - (10*x+4*x^2)*d/dx P(n-2,x) - 2*x^2*d^2/dx^2 P(n-2,x) + (6+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(8 + 10*x + 2*x^2)*P[n - 2, x] - (10*x + 4*x^2)*D[P[n - 2, x], x] - 2*x^2*D[P[n - 2, x], {x, 2}] + (6 + 3*x)*P[n - 1, x] + 3*x*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; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 10*x + 2*x^2)*v[i - 2] - (10*x + 4*x^2)*deriv(v[i - 2], x) - 2*x^2*deriv(deriv(v[i - 2], x), x) + (6 + 3*x)*v[i - 1] + 3*x*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 (−∞, 0]
by Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? 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

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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741