A123097 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(8 + 24\,x + 18\,x^{2} - 4\,n - 12\,n x - 9\,n x^{2}\right)\,P_{n-2}(x) + \left(-8 - 12\,x + 4\,n + 6\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 6\,x\)

\(P_{2}(x) = 14 + 39\,x + 27\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	6														11
2	14	39	27													80
3	36	156	225	108												525
4	88	516	1134	1107	405											3250
5	208	1536	4536	6696	4941	1458										19375
6	480	4272	15840	31320	34830	20655	5103									112500
7	1088	11328	50544	125280	186300	166212	82377	17496								640625
8	2432	28992	151200	450576	839160	1000188	745038	317115	59049							3593750
9	5376	72192	430848	1499904	3356640	5007744	4980528	3184272	1187541	196830						19921875
10	11776	175872	1181952	4707072	12301632	22044960	27433728	23409648	13108878	4349943	649539					109375000
11	25600	420864	3144960	14100480	42145920	88179840	131779872	140667840	105107220	52356780	15647985	2125764				595703125
12	55296	992256	8160768	40677120	136857600	327431808	571209408	732102624	684181080	454677300	203955246	55447011	6908733			3222656250
13	118784	2310144	20736000	113743872	425399040	1145498112	2284837632	3418018560	3834878256	3187071360	1907046504	778029624	193975965	22320522		17333984375
14	253952	5320704	51757056	309823488	1275056640	3816274176	8566601472	14651290368	19184783904	19139749200	14321035872	7793050824	2915485326	671209983	71744535	92773437500

Technical details

Node id: n-f9a27f3541c50e99 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-d1e401f52851d799

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:    5,     6;
  2:   14,    39,     27;
  3:   36,   156,    225,     108;
  4:   88,   516,   1134,    1107,     405;
  5:  208,  1536,   4536,    6696,    4941,    1458;
  6:  480,  4272,  15840,   31320,   34830,   20655,    5103;
  7: 1088, 11328,  50544,  125280,  186300,  166212,   82377,   17496;
  8: 2432, 28992, 151200,  450576,  839160, 1000188,  745038,  317115,   59049;
  9: 5376, 72192, 430848, 1499904, 3356640, 5007744, 4980528, 3184272, 1187541, 196830;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(12*n-24)*x+(9*n-18)*x^2)*P(n-2,x) + (4*n-8+(6*n-12)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 6*x, P(2,x) = 14 + 39*x + 27*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 6*x;
P[2, x_] := 14 + 39*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (12*n - 24)*x + (9*n - 18)*x^2)*P[n - 2, x] + (4*n - 8 + (6*n - 12)*x)*P[n - 1, 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] = 1; v[2] = 5 + 6*x; v[3] = 14 + 39*x + 27*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (12*n - 24)*x + (9*n - 18)*x^2)*v[i - 2] + (4*n - 8 + (6*n - 12)*x)*v[i - 1])/(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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) = binomial(n-2, k-1) + n*binomial(n-1, k-1), 1 <= k <= n, starting with T(1, 1) = 1.?
  - 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