A316649 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + n^{2}\right)\,P_{n}(x) = \left(-8 - 10\,x - 2\,x^{2} + 12\,n + 15\,n x + 3\,n x^{2} - 4\,n^{2} - 5\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-10\,x - 4\,x^{2} + 15\,n x + 6\,n x^{2} - 5\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} + 3\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(12 + 18\,x + 4\,x^{2} - 30\,n - 45\,n x - 10\,n x^{2} + 12\,n^{2} + 18\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(18\,x + 8\,x^{2} - 45\,n x - 20\,n x^{2} + 18\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n-1}'(x) + \left(4\,x^{2} - 10\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 3 + 7\,x + 2\,x^{2}\)

\(P_{2}(x) = 26 + 196\,x + 229\,x^{2} + 72\,x^{3} + 6\,x^{4}\)

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	7	2													12
2	26	196	229	72	6											529
3	252	5238	14964	12525	3942	490	20									37431
4	2568	140046	849486	1446813	967041	294220	42860	2870	70							3745974
5	26928	3784122	45879342	143327859	172396431	96848383	28087334	4394810	369362	15498	252					495130321
6	287648	103446622	2433280978	13231699853	26503333389	24560881196	11953669939	3286092558	529675608	50437800	2760366	79464	924			82655646345
7	3112896	2857227954	128377246062	1179569964895	3766915267031	5424335669871	4082246816362	1756737648109	456973363004	74162294136	7596757056	486477222	18722748	392964	3432	16880280963742

All 15 rows

Technical details

Node id: n-80eea50c2afaf945 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-0ea4d4f1315e31da

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,             7,               2;
  2:        26,           196,             229,               72,                 6;
  3:       252,          5238,           14964,            12525,              3942,                490,                 20;
  4:      2568,        140046,          849486,          1446813,            967041,             294220,              42860,               2870,                 70;
  5:     26928,       3784122,        45879342,        143327859,         172396431,           96848383,           28087334,            4394810,             369362,             15498,               252;
  6:    287648,     103446622,      2433280978,      13231699853,       26503333389,        24560881196,        11953669939,         3286092558,          529675608,          50437800,           2760366,            79464,             924;
  7:   3112896,    2857227954,    128377246062,    1179569964895,     3766915267031,      5424335669871,      4082246816362,      1756737648109,       456973363004,       74162294136,        7596757056,        486477222,        18722748,         392964,         3432;
  8:  34013312,   79596165718,   6774762227050,  103320259073289,   512118407101865,   1105158713246224,   1226564760483055,    776462189904247,    299867423103421,    73821188886636,    11900589122472,    1272391447866,     90096809418,     4145701560,    118346514,     1891890,     12870;
  9: 374416128, 2232931262922, 358420712443062, 8972825636866539, 67835869600154691, 214257356209845907, 340186246320531770, 304835492952661454, 166444937238088711, 58348944732306135, 13608912548141990, 2161144838184002, 236645734622902, 17912626417686, 928891953132, 32196184020, 708975410, 8921770, 48620;

%F Formula

The row polynomials satisfy (n^2-2*n)*P(n,x) = -(4*n^2-12*n+8+(5*n^2-15*n+10)*x+(n^2-3*n+2)*x^2)*P(n-2,x) - ((5*n^2-15*n+10)*x+(2*n^2-6*n+4)*x^2)*d/dx P(n-2,x) - (n^2-3*n+2)*x^2*d^2/dx^2 P(n-2,x) + (12*n^2-30*n+12+(18*n^2-45*n+18)*x+(4*n^2-10*n+4)*x^2)*P(n-1,x) + ((18*n^2-45*n+18)*x+(8*n^2-20*n+8)*x^2)*d/dx P(n-1,x) + (4*n^2-10*n+4)*x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 7*x + 2*x^2, P(2,x) = 26 + 196*x + 229*x^2 + 72*x^3 + 6*x^4.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 7*x + 2*x^2;
P[2, x_] := 26 + 196*x + 229*x^2 + 72*x^3 + 6*x^4;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 12*n + 8 + (5*n^2 - 15*n + 10)*x + (n^2 - 3*n + 2)*x^2)*P[n - 2, x] - ((5*n^2 - 15*n + 10)*x + (2*n^2 - 6*n + 4)*x^2)*D[P[n - 2, x], x] - (n^2 - 3*n + 2)*x^2*D[P[n - 2, x], {x, 2}] + (12*n^2 - 30*n + 12 + (18*n^2 - 45*n + 18)*x + (4*n^2 - 10*n + 4)*x^2)*P[n - 1, x] + ((18*n^2 - 45*n + 18)*x + (8*n^2 - 20*n + 8)*x^2)*D[P[n - 1, x], x] + (4*n^2 - 10*n + 4)*x^2*D[P[n - 1, x], {x, 2}])/(n^2 - 2*n)]];
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 + 7*x + 2*x^2; v[3] = 26 + 196*x + 229*x^2 + 72*x^3 + 6*x^4; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 12*n + 8 + (5*n^2 - 15*n + 10)*x + (n^2 - 3*n + 2)*x^2)*v[i - 2] - ((5*n^2 - 15*n + 10)*x + (2*n^2 - 6*n + 4)*x^2)*deriv(v[i - 2], x) - (n^2 - 3*n + 2)*x^2*deriv(deriv(v[i - 2], x), x) + (12*n^2 - 30*n + 12 + (18*n^2 - 45*n + 18)*x + (4*n^2 - 10*n + 4)*x^2)*v[i - 1] + ((18*n^2 - 45*n + 18)*x + (8*n^2 - 20*n + 8)*x^2)*deriv(v[i - 1], x) + (4*n^2 - 10*n + 4)*x^2*deriv(deriv(v[i - 1], x), x))/(n^2 - 2*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 (−∞, 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 read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] order…?
  - 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