A123097 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(8 + 16\,x + 8\,x^{2} - 4\,n - 8\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(-8 - 8\,x + 4\,n + 4\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 4\,x\)

\(P_{2}(x) = 14 + 26\,x + 12\,x^{2}\)

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	4														9
2	14	26	12													52
3	36	104	100	32												272
4	88	344	504	328	80											1344
5	208	1024	2016	1984	976	192										6400
6	480	2848	7040	9280	6880	2720	448									29696
7	1088	7552	22464	37120	36800	21888	7232	1024								135168
8	2432	19328	67200	133504	165760	131712	65408	18560	2304							606208
9	5376	48128	191488	444416	663040	659456	437248	186368	46336	5120						2686976
10	11776	117248	525312	1394688	2429952	2903040	2408448	1370112	511488	113152	11264					11796480
11	25600	280576	1397760	4177920	8325120	11612160	11569152	8232960	4101120	1361920	271360	24576				51380224
12	55296	661504	3627008	12052480	27033600	43118592	50147328	42848256	26695680	11827200	3536896	641024	53248			222298112
13	118784	1540096	9216000	33701888	84029440	150847488	200589312	200048640	149630976	82903040	33071104	8994816	1495040	114688		956301312
14	253952	3547136	23003136	91799552	251863040	502554624	752074752	857505792	748560384	497868800	248348672	90095616	22470656	3448832	245760	4093640704

Technical details

Node id: n-3d01f9226121973c · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-36dd966291f10a09

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,     4;
  2:   14,    26,     12;
  3:   36,   104,    100,     32;
  4:   88,   344,    504,    328,     80;
  5:  208,  1024,   2016,   1984,    976,    192;
  6:  480,  2848,   7040,   9280,   6880,   2720,    448;
  7: 1088,  7552,  22464,  37120,  36800,  21888,   7232,   1024;
  8: 2432, 19328,  67200, 133504, 165760, 131712,  65408,  18560,  2304;
  9: 5376, 48128, 191488, 444416, 663040, 659456, 437248, 186368, 46336, 5120;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(8*n-16)*x+(4*n-8)*x^2)*P(n-2,x) + (4*n-8+(4*n-8)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 4*x, P(2,x) = 14 + 26*x + 12*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 4*x;
P[2, x_] := 14 + 26*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (8*n - 16)*x + (4*n - 8)*x^2)*P[n - 2, x] + (4*n - 8 + (4*n - 8)*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 + 4*x; v[3] = 14 + 26*x + 12*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (8*n - 16)*x + (4*n - 8)*x^2)*v[i - 2] + (4*n - 8 + (4*n - 8)*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