A153648 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-30 - 30\,x + 15\,n + 15\,n x\right)\,P_{n-2}(x) + \left(-30\,x + 15\,n x\right)\,P_{n-2}'(x) + \left(-4 - 2\,x + 2\,n + n x\right)\,P_{n-1}(x) + \left(-2\,x + n x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 10 + 5\,x\)

\(P_{2}(x) = 50 + 52\,x + 2\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	10	5														15
2	50	52	2													104
3	250	506	135	2												893
4	1250	4078	1916	175	2											7421
5	6250	32414	25407	4936	217	2										69226
6	31250	244582	281432	89327	9013	261	2									655867
7	156250	1831166	2999835	1405332	233720	14275	307	2								6640887
8	781250	13455958	30163676	19607595	4823532	492330	20856	355	2							69345554
9	3906250	98427854	296570727	257519096	87157767	13060392	905538	28896	405	2						757576927
10	19531250	714587302	2843915552	3213077047	1436344523	295243191	29879526	1521042	38541	457	2					8554138433
11	97656250	5174722526	26912350035	38809007452	22230763150	5985848645	825266769	60619494	2394567	49943	511	2				100098679344
12	488281250	37352411638	251519132036	456400743395	328115581282	112248758700	20153980892	2001560145	112583829	3590605	63260	567	2			1208396687601
13	2441406250	269252035694	2332105529247	5265748546656	4674536579117	1986044717482	449921346256	57821363012	4359957390	195490439	5183185	78656	625	2		15042432234011
14	12207031250	1938094081222	21476321268872	59817679846767	64767647768433	33600971603721	9385314861690	1512810544144	146643156002	8737700404	321985634	7256673	96301	685	2	192666757201800

Technical details

Node id: n-6443a135cc2a3988 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-0965ba4bf8416ad4

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:       2;
  1:      10,        5;
  2:      50,       52,         2;
  3:     250,      506,       135,         2;
  4:    1250,     4078,      1916,       175,        2;
  5:    6250,    32414,     25407,      4936,      217,        2;
  6:   31250,   244582,    281432,     89327,     9013,      261,      2;
  7:  156250,  1831166,   2999835,   1405332,   233720,    14275,    307,     2;
  8:  781250, 13455958,  30163676,  19607595,  4823532,   492330,  20856,   355,   2;
  9: 3906250, 98427854, 296570727, 257519096, 87157767, 13060392, 905538, 28896, 405, 2;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (15*n-30+(15*n-30)*x)*P(n-2,x) + (15*n-30)*x*d/dx P(n-2,x) + (2*n-4+(n-2)*x)*P(n-1,x) + (n-2)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 10 + 5*x, P(2,x) = 50 + 52*x + 2*x^2.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 10 + 5*x;
P[2, x_] := 50 + 52*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((15*n - 30 + (15*n - 30)*x)*P[n - 2, x] + (15*n - 30)*x*D[P[n - 2, x], x] + (2*n - 4 + (n - 2)*x)*P[n - 1, x] + (n - 2)*x*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 10 + 5*x; v[3] = 50 + 52*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((15*n - 30 + (15*n - 30)*x)*v[i - 2] + (15*n - 30)*x*deriv(v[i - 2], x) + (2*n - 4 + (n - 2)*x)*v[i - 1] + (n - 2)*x*deriv(v[i - 1], x))/(n - 2)); 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 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + j*prime(j)*T(n-2, k-1) with j=3, read by rows.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? 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