A153518 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(-15 - 15\,x + 5\,n + 5\,n x\right)\,P_{n-2}(x) + \left(-15\,x + 5\,n x\right)\,P_{n-2}'(x) + \left(-6 - 3\,x + 2\,n + n x\right)\,P_{n-1}(x) + \left(-3\,x + n x\right)\,P_{n-1}'(x)\)

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

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

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

\(P_{3}(x) = 250 + 506\,x + 135\,x^{2} + 2\,x^{3}\)

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	750	2538	1336	155	2											4781
5	2750	14674	12437	2826	177	2										32866
6	9250	75902	97152	36347	4713	201	2									223567
7	32250	397446	724435	397592	83180	7065	227	2								1642197
8	110750	2029858	5131976	3925095	1196232	162230	9956	255	2							12566354
9	382750	10336034	35411517	36331466	15169947	2959692	285148	13466	285	2						100890307
10	1319250	52243182	239111032	321230627	176882423	46735851	6400486	466322	17681	317	2					844407173
11	4552250	263322886	1591540235	2748951072	1943431170	668673875	122718379	12561764	723287	22693	351	2				7356497964
12	15700750	1323548978	10477365216	22956463295	20437751802	8910635940	2108117172	286429565	22921889	1077165	28600	387	2			66540040761
13	54162750	6642337794	68422727797	188196404306	207913508717	112589575482	33414085956	5802045712	611005190	39521789	1553135	35506	425	2		623686964561
14	186829250	33295169662	444111472112	1520920841307	2061403568133	1365548374301	498239543850	107630265824	14375730442	1214212564	65118234	2180933	43521	465	2	6046993350600

Technical details

Node id: n-5f02b0c44d7cf474 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-24712543c7b52d43

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:    750,     2538,     1336,      155,        2;
  5:   2750,    14674,    12437,     2826,      177,       2;
  6:   9250,    75902,    97152,    36347,     4713,     201,      2;
  7:  32250,   397446,   724435,   397592,    83180,    7065,    227,     2;
  8: 110750,  2029858,  5131976,  3925095,  1196232,  162230,   9956,   255,   2;
  9: 382750, 10336034, 35411517, 36331466, 15169947, 2959692, 285148, 13466, 285, 2;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (5*n-15+(5*n-15)*x)*P(n-2,x) + (5*n-15)*x*d/dx P(n-2,x) + (2*n-6+(n-3)*x)*P(n-1,x) + (n-3)*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 2, P(1,x) = 10 + 5*x, P(2,x) = 50 + 52*x + 2*x^2, P(3,x) = 250 + 506*x + 135*x^2 + 2*x^3.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 10 + 5*x;
P[2, x_] := 50 + 52*x + 2*x^2;
P[3, x_] := 250 + 506*x + 135*x^2 + 2*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((5*n - 15 + (5*n - 15)*x)*P[n - 2, x] + (5*n - 15)*x*D[P[n - 2, x], x] + (2*n - 6 + (n - 3)*x)*P[n - 1, x] + (n - 3)*x*D[P[n - 1, x], x])/(n - 3)]];
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; v[4] = 250 + 506*x + 135*x^2 + 2*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((5*n - 15 + (5*n - 15)*x)*v[i - 2] + (5*n - 15)*x*deriv(v[i - 2], x) + (2*n - 6 + (n - 3)*x)*v[i - 1] + (n - 3)*x*deriv(v[i - 1], x))/(n - 3)); 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 Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.?
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - 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