A134239 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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

\(P_{2}(x) = 18 + 45\,x + 27\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	6	6							12
2	18	45	27						90
3	48	192	252	108					600
4	120	660	1350	1215	405				3750
5	288	2016	5616	7776	5346	1458			22500
6	672	5712	20160	37800	39690	22113	5103		131250
7	1536	15360	65664	155520	220320	186624	87480	17496	750000

All 15 rows

Technical details

Node id: n-fd95b73866112a83 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-b821bbf6b18e27b2

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,     6;
  2:   18,    45,     27;
  3:   48,   192,    252,     108;
  4:  120,   660,   1350,    1215,     405;
  5:  288,  2016,   5616,    7776,    5346,    1458;
  6:  672,  5712,  20160,   37800,   39690,   22113,    5103;
  7: 1536, 15360,  65664,  155520,  220320,  186624,   87480,   17496;
  8: 3456, 39744, 199584,  571536, 1020600, 1163484,  826686,  334611,   59049;
  9: 7680, 99840, 576000, 1935360, 4173120, 5987520, 5715360, 3499200, 1246590, 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) = 6 + 6*x, P(2,x) = 18 + 45*x + 27*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 6*x;
P[2, x_] := 18 + 45*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] = 6 + 6*x; v[3] = 18 + 45*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)
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.