A134239 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 6 + 4\,x\)

\(P_{2}(x) = 18 + 30\,x + 12\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	4														10
2	18	30	12													60
3	48	128	112	32												320
4	120	440	600	360	80											1600
5	288	1344	2496	2304	1056	192										7680
6	672	3808	8960	11200	7840	2912	448									35840
7	1536	10240	29184	46080	43520	24576	7680	1024								163840
8	3456	26496	88704	169344	201600	153216	72576	19584	2304							737280
9	7680	66560	256000	573440	824320	788480	501760	204800	48640	5120						3276800
10	16896	163328	709632	1824768	3075072	3548160	2838528	1554432	557568	118272	11264					14417920
11	36864	393216	1904640	5529600	10690560	14450688	13934592	9584640	4608000	1474560	282624	24576				62914560
12	79872	931840	4978688	16107520	35143680	54472704	61501440	50958336	30750720	13178880	3807232	665600	53248			272629760
13	172032	2179072	12730368	45416448	110387200	193019904	249790464	242221056	175988736	94617600	36585472	9633792	1548288	114688		1174405120
14	368640	5038080	31948800	124600320	333864960	650158080	948879360	1054310400	896163840	579870720	281149440	99041280	23961600	3563520	245760	5033164800

Technical details

Node id: n-ed2a76556fde8ee6 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-8a8f26bfa265b5f7

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:    6,     4;
  2:   18,    30,     12;
  3:   48,   128,    112,     32;
  4:  120,   440,    600,    360,     80;
  5:  288,  1344,   2496,   2304,   1056,    192;
  6:  672,  3808,   8960,  11200,   7840,   2912,    448;
  7: 1536, 10240,  29184,  46080,  43520,  24576,   7680,   1024;
  8: 3456, 26496,  88704, 169344, 201600, 153216,  72576,  19584,  2304;
  9: 7680, 66560, 256000, 573440, 824320, 788480, 501760, 204800, 48640, 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) = 6 + 4*x, P(2,x) = 18 + 30*x + 12*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 4*x;
P[2, x_] := 18 + 30*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] = 6 + 4*x; v[3] = 18 + 30*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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from A127899(unsigned) * A007318.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? 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

Related ?

Same fingerprint ?

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.